What really is a colimit of sets?

Question

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!

Answer 1

Limits and colimits don't coincide in $Set$. Just like any colimit, it suffices to understand coproducts and coequalizers. A coproduct of a family $\{A_i\}_{i\in I}$ of sets is their disjoint union. More formally, take $$X=\bigcup_{\{i\in I\}}\{i\}\times A_i$$with the obvious injections $A_i\to X$. Then $X$ is a coproduct. Let me know if you need help figuring out coequalizers.

Answer 2

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.

Answer 3

Here is another perspective on colimits that I find useful, using some intuition from modules and tensor products.

Let $D$ be your diagram category. A left $D$-module is a functor $M:D\rightarrow Set$. If $\gamma:i\rightarrow j$ is a morphism in $D$ then we will denote the associated morphism $M(i)\rightarrow M(j)$ by $m\mapsto \gamma\cdot m$. Similarly, a right $D$-module is a functor $D^{op}\rightarrow Set$, and we denote the "action" of $\gamma$ as right multiplication.

Given a right $D$-module $N$ and a left $D$-module $M$ we may form $$ N\times_D M := \left(\bigsqcup_i N_i\times M_i\right)\bigg/\sim $$ where $\sim$ identifies $(n\cdot \gamma,m)\sim (n,\gamma\cdot m)$ for all $n\in N(j)$, $m\in M(i)$ and all edges $\gamma:i\rightarrow j$.

Now, the colimit of a functor $M:D\rightarrow Set$ is just given by "tensoring" with the trivial module: $$ \mathrm{colim}(M) = \mathrm{triv}\times_D M $$ where the trivial module is defined by $\mathrm{triv}(i)=\{\mathrm{pt}\}$ for all $i$, and each $\gamma:\mathrm{triv}(i)\rightarrow \mathrm{triv}(j)$ is the identity.

For an example, if $D$ is the category with two objects 1,2 and two morphisms $a,b:1\rightarrow 2$, then a left $D$-module is a choice of sets $M_1,M_2$ and two morphisms $a,b:M_1\rightarrow M_2$. The colimit of this $M$ is $$ \Big(\{\mathrm{pt}\}\times M_1 \ \sqcup \ \{\mathrm{pt}\}\times M_2\Big)\Big/\sim $$ where we identify $(\mathrm{pt},m) = (\mathrm{pt}\cdot a,m) = (\mathrm{pt},a\cdot m)$ and $(\mathrm{pt},m) = (\mathrm{pt}\cdot b,m) = (\mathrm{pt},b\cdot m)$ for all $m\in M_1$. Clearly, this is isomorphic to $M_2 / \langle a(m)\sim b(m)\:|\: \forall m\in M_1\rangle$. This is the usual coequalizer construction in $Set$.

I like this a lot because it is so easy to generalize to other settings. For instance, you can construct homotopy colimits (!) in a category of chain complexes of $\mathbb{k}$-modules as $$ \mathbf{P}(\mathrm{triv})\otimes_{\mathbb{k}[D]} M, $$ where $\mathbb{k}[D]$ denotes the linearization of $D$ (could be thought of as the path algebra of a quiver), and $\mathbf{P}(\mathrm{triv})$ denotes a projective resolution of its trivial right module.