With Respect To
Definitions
Most mathematical expressions, in order to be evaluated, require a large amount of context. For instance, if I define let x equal 1+2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \t{let } x \t{ equal } 1 + 2 then in order to know what x %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x is we first have to know what 1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 1 is, what 2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 2 is, what + %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } + is, and what notion of “equality” we are using1.
These might seem silly—obviously 1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 1 is one and 2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 2 is two and + %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } + is addition (and everyone knows what equality is). But perhaps we’re working in a modular group Z/p %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \bb Z / p where 1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 1 and 2 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 2 are not the usual numbers but equivalence classes thereof? Or maybe we’re working in some model where the symbol 1 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 1 is actually interpreted as the number 100 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 100 (this could come about by considering the image of (Z,+) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (\bb Z, +) under the automorphism x↦x+99 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x \mapsto x + 99.
In a similar vein, a set X %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } X cannot be considered ‘open’ or ‘closed’ until you tell me in which topology X %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } X is being considered; objects a %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a and b %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } b cannot be isomorphic but must be isomorphic in some category; “the group Z %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \bb Z" is ambiguous until you specify a group operation; the expression 1/n %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 1/n may or may not be well-founded depending on whether we are working in R %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \bb R or Z %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \bb Z; so on, so on.
Definition
If we have some expressoin/statement/formula/symbol A %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } A whose meaning depends on some other thing (or ‘parameter’) B %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B, then we may say that A %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } A is defined with respect to (wrt)2 B %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B or relative to B %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } B.
As a side-note: we could read the phrase “with respect to” as literally meaning “giving respect to”, which makes some sense. When we say “XYZ with respect to A”, we are in some literal sense “giving respect to” A in our interpretation of XYZ.
Often times these relationships are explicit; in the definition p(x,y)=2x+3y %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } p(x, y) = 2x + 3y the value p(x,y) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } p(x, y) is defined relative to the values of x %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x and y %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } y, and this is reflected by the notation. This is not always the case, however, and is in particular not the case when the parameters with respect to which we perform a construction do not vary throughout the course of a discussion. So although the expression 7+5 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 7 + 5 has very different meanings depending on whether + %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } + is addition in (Z,+) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (\bb Z, +) versus (Z/p,+) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } (\bb Z / p, +), most of the time we will not bother to specify 7+Z5or7+Z/p5 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } 7 +_\bb Z 5 \hspace{10pt}\t{or}\hspace{10pt} 7+_{\bb Z/p} 5 and instead rely on context for disambiguation. (Otherwise notation would get so cluttered!) Likewise, although a set U %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } U can only be considered ‘open’ relative to some topology τ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \tau, it’s usually3 not worth repeating “with respect to τ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \tau" throughout our discussion every time we want to call a set open. Instead we establish early on which topology we are working with and then proceed to simply call a set “open” or “closed”.
One time it is worth the disambiguation is when we are working with two different topologies τ,ρ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \tau, \rho. Then context only tells us that a set U %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } U being ‘open’ means it is probably open in either τ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \tau or ρ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \rho, but not which.