Let
C be some
category, and let
A∈C. Our goal is to define a
category CA whose
objects consist exactly of those
morphisms in
C into
A. That is,
f∈Obj(CA) exactly when f:Z→A within
C for some
Z∈C.
The book then poses me this question: what should the
morphisms be to turn this into a
category?
There are of course the trivial solutions, like using a discrete
category. I’m not positive, but I think it’d also work to define
Hom(f:X→CA,g:Y→CA)=HomC(X,Y)
So let’s see if that does work.
Given
f∈Obj(CA) a
morphism X→CA we do indeed have at
least one
morphism in
HomCA(f,f); namely,
1X∈HomC(X,X). (Also have to show that these
act as identities, though)
We can compose using composition from
C. If
α∈HomCA(f,g) and
β∈HomCA(g,h) then we know
f:X→CAg:Y→CAh:Z→CAα:X→CYβ:Y→CZfor some Xfor some Yfor some Z
so using the composition from
C we get
β∘α:X→CZ meaning that indeed
β∘α∈HomCA(f,h)
Is composition
associative? Since it’s just composition from
C, it must be
Likewise, we get that the chosen identities are indeed identities.
... Well, the book uses a stricter definition, defining a
morphism to be a commuting triangle. (What you get is called a
slice category.) But according to
this math.SE question, my definition also forms a
category (yay!). Just not an interesting one.
and morphisms are commutative diagrams
The book would like me to define composition of morphisms in this category and verify that the result satisfies the category laws.
Composition and identities are pulled directly from the ambient category: given morphisms in , their composition is ; given an object out of , its identity is the -morphism .
Now we verify that the category laws are obeyed.