I am attempting to do a calculation of a colimit in $Cat$, the category of small categories. To this end, people have suggested that I do this by calculating coproducts and using coequalizers. I have no idea how to do this. I have seen the definition for coproducts in Cat but how do we then use coequalizers to find colimits? I am going to have an infinite diagram over which I want to take a colimit to find a particular category. I want to build up my colimit diagram by having a small number of objects in $Cat$, then adjoining one new object to this and finding the new colimit. I can imagine taking successive coproducts, but I am really shakey as to how to do all this. Can anyone give me some direction?
[Math] colimits in Cat via coproducts and coequalizers
ct.category-theorylimits-and-colimits
Best Answer
Firstly, there exists a general method to construct colimits in arbitrary category via coproducts and coequalizers. I will point it briefly. Let $A$ and $B$ be categories, $T\colon A\to B$ be a functor. If $B$ has coproducts of all families indexed by objects and morphisms of $A$ and all binary coequalizers, then such colimit exists. It is the coequalizer of the pair $(F,G)$, where $F$ and $G$ are morphisms from $\coprod_{f\in Arr(A)}dom(T(f))$ to $\coprod_{a\in Ob(A)}T(a)$, such that $F\circ i_f=i_{dom(f)}$ and $G\circ i_f=i_{cod(f)}\circ T(f)$. This fact is dual to the analogous one about constructing limits via products and equalizers, which you can find in CFWM. It is a general picture.
The category $\mathbf{Cat}$ is cocomplete, i.e for every small category (graph) $A$ and every functor (diagram) $T\colon A\to\mathbf{Cat}$ there exists a colimit of $T$, which one can construct via corresponding coproducts and coequalizers in $\mathbf{Cat}$.
It may be difficult to calculate coequalizers in $\mathbf{Cat}$. Let $A$ and $B$ be categories, $T,S\colon A\to B$ be functors between them. Then the coequalizer of the pair $(T,S)$ is $B/C$, where $C$ is the free congruence generated by $T(a)=S(a)$ and $T(f)=S(f)$ for any object $a\in A$ and any morphism $f\in A$. See also: Generalized congruences -- Epimorphisms in Cat.