[Math] What really is a colimit of sets

category-theorylimits-colimits

This is probably a question I should have asked myself a bit earlier. For some reason I always thought I knew the answer so I did not bother, but now that I actually need to use it (I am studying the $+$ construction on presheaves) I realize I am not really familiar with this.

So, what is a colimit of sets, formally?! Let $F:\mathcal{D}\to \mathbf{Set}$ be a diagram in $\mathbf{Set}$. What is $\mathrm{Colim}\;F$ ?

I believe (am I right?) that if $\mathcal{D}$ is filtered, then the colimit coincides with the direct limit, but what is it in the general case? Thanks!

Best Answer

To make my answer clearer, firstly I describe some basic concepts in category theory.

  1. Suppose you have two categories $A$ and $B$ and functor $T\colon A\to B$. Then we can take its limit $\varprojlim T$ and its colimit $\varinjlim T$(see definitions here: limit and colimit). They may not exist(or one of them may not exists), but if they exist, then they are objects in $B$, defined up to isomorphism. Note that a colimit of $T$ is nothing but a limit of the dual functor $T^{op}\colon A^{op}\to B^{op}$.
  2. Suppose you have a graph $D$, a category $B$ and a diagram $F\colon D\to B$. Then you can take a limit(respectively, colimit) of the diagram $F$, which is nothing but a limit(respectively, colimit) of the corresponding functor $C[F]\colon C[D]\to B$, where $C[D]$ is a free category on the grath $D$.
  3. Well, now we can take limits and colimits of $\mathbf{Set}$-valued functors $T\colon A\to\mathbf{Set}$ and $\mathbf{Set}$-valued diagrams $F\colon D\to\mathbf{Set}$. From 2 we understand that we can reduce our problem to (co)limits of $\mathbf{Set}$-valued functors. There is an important result about such (co)limits: Theorem. Let $A$ be a small category and $T\colon A\to\mathbf{Set}$ be a functor. Then the colimit(respectively, limit) of $T$ exists and given by a formula(in the case of colimit): $$ \varinjlim T=\coprod_{a\in A}T(a)/\sim $$ where $\sim$ is the minimal equivalence relation on $\coprod_{a\in A}T(a)$, contains all pairs $((a,x),(a',x'))\in\coprod_{a\in A}T(a)\times\coprod_{a\in A}T(a)$, such that there exists a morphism $f\in Arr(A)$, $f\colon a\to a'$ satisfying $(T(f))(x)=x'$. The proof is straightforward.
  4. Direct limit in modern mathematics is nothing but a colimit(but in some literature you can find this term in the meaning of directed colimit, see 5).
  5. There is a notion of directed colimit which is a colimit of a functor from a preorder, corresponding to some directed set. This is, of course, the special case of the aforementioned construction.
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