Cones

Definition

A cone over a diagram $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F : J \to C$ with summit $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } c \in C$ is a natural transformation $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \lambda : \Delta(c) \Rightarrow F$ from the constant functor $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \Delta(c)$ to $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F$. Each component $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \lambda_j$ is called a leg. The naturality square in this case looks like: which is the same as The triangular shape of this commutative diagram gives rise to the name “cone” for the object being discussed

Dualization

Dually, a cone under $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F$ (or co-cones) with nadir $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } c \in C$ is a natural transformation $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \mu : F \Rightarrow \Delta(c)$.

Cone Category

For any diagram $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F : J \to C$, we can create the cone functor $$\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 } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \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} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \begin{align*} \mathrm{Cone}(-, F) &: C^\mathrm{op} \to \cat{{Set}} \\ &\phantom{:} c &&\mapsto \mathrm{Nat}(\Delta(c), F) = \t{cones over } F \t{ with summit } c \\ &\phantom{:} (f : c \to c') &&\mapsto (\mu \mapsto \eta \t{ given by } \eta_j = \mu_j f) \end{align*}$$ Then the category of elements $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \int \text{Cone}(-, F)$ gives rise to the category of cones over $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F$ whose objects are cones over $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } \newcommand{\box}[1]{ \fbox{$ #1 $} } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathrel{↤} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F$ and whose morphisms are commutative diagrams