# How to construct colimits in the category of diagrams in $\mathbf{Sets}$

Let $$\mathbf{DSets}$$ be the category with

• Functors $$F\colon \mathbf{C} \to \mathbf{Sets}$$ as objects, where $$\mathbf{C}$$ is a category,
• Pairs $$(L,\lambda)$$ as morphisms between objects $$F \colon \mathbf{C} \to \mathbf{Sets}$$ and $$F' \colon \mathbf{C'} \to \mathbf{Sets}$$, where $$L$$ is a functor from $$\mathbf{C}$$ to $$\mathbf{C'}$$, and $$\lambda$$ is a natural transformation from $$F$$ to $$F'L$$.

I've been informed that $$\mathbf{DSets}$$ is cocomplete, and I'm trying to understand how colimits are built. Consider the small example below.

• Let $$\mathbf{C}$$ be the category with two objects $$X$$ and $$Y$$ and only one non-trivial morphism $$f$$ between them.

• Let $$F\colon \mathbf{C} \to \mathbf{Sets}$$ be the functor such that $$F(X) = \{a,b\}$$, $$F(Y)=\{c,d,e\}$$, $$F(f)(a)=c$$, and $$F(f)(b)=d$$.

• Let $$F'\colon \mathbf{C} \to \mathbf{Sets}$$ be the functor such that $$F'(X) = \{p,q,r\}$$, $$F(Y)=\{u,v\}$$, $$F(f)(p)=u$$, $$F(f)(q)=v$$, and $$F(f)(r)=v$$.

Let $$T$$ be the terminal object of $$\mathbf{DSets}$$, and let

• $$(L,\lambda)$$ be the morphism from $$T$$ to $$F$$ such that the single object of $$T$$ is sent to $$Y$$, and the singleton is sent to $$c$$, and
• $$(L',\lambda')$$ be the morphism from $$T$$ to $$F'$$ such that the single object of $$T$$ is sent to $$X$$, and the singleton is sent to $$r$$.

What would be the pushout $$R \colon \mathbf{C''} \to \mathbf{Sets}$$ of the diagram $$F \leftarrow T \rightarrow F'$$ ? What I have so far:

• $$\mathbf{C''}$$ is the category with three objects $$X'$$, $$Y'$$ and $$Z'$$ and three non-trivial morphisms $$X' \to Y'$$, $$Y' \to Z'$$, $$X' \to Z'$$. It comes from amalgamating $$Y$$ and $$X$$ in $$\mathbf{C}$$ and $$\mathbf{C'}$$.
• $$R(X')$$ is a two-element set $$\{a',b'\}$$, $$R(Z')$$ is a two-element set $$\{u',v'\}$$, and $$R(Y')$$ is a five-element set $$\{c|r,d',e',p',q'\}$$ which is the amalgamated coproduct of $$F(Y)$$ and $$F'(X)$$,
• $$R(X' \to Y')$$ sends $$a'$$ and $$b'$$ to $$c|r$$ and $$d'$$ respectively, and $$R(Y' \to Z')$$ sends $$c|r$$, $$q'$$ and $$p'$$ to $$v$$, $$v$$, and $$u$$ respectively.

To finish the construction of this colimit, I should identify the images of $$d'$$ and $$e'$$ by $$R(Y' \to Z')$$, but this is where I'm confused because I don't see how I should pick them, and the definition of colimits is not helping me.

Rather than giving the solution directly, I would appreciate any hint to find it myself.

$$\mathbf{DSets}$$ admits a canonical forgetful functor $$\pi$$ to the category $$\mathbf{Cat}$$ of small categories. Given a diagram $$D:J\to \mathbf{DSets}$$, its colimit is a diagram indexed by the colimit, call in $$\mathbf{C}_D$$, of $$\pi\circ D$$. To construct it, we first construct a diagram $$D':J\to [\mathbf{C}_D,\mathbf{Set}]$$ via the left Kan extension along the legs $$\pi(D(j))\to \mathbf{C}_D$$ of the cocone defining $$\mathbf{C}_D$$. Then $$\mathrm{colim} D=\mathrm{colim} D'$$.
For instance, in your example we have $$J=\bullet\leftarrow\bullet\to \bullet$$ and $$\pi \circ D =(X\to Y)\leftarrow Y \to (Y\to Z)$$, while $$\mathrm{colim}\pi\circ D= X\to Y\to Z$$. The left Kan extension along the inclusion $$(X\to Y)\to (X\to Y\to Z)$$ sends a functor $$F=S\to T$$, where $$S=F(X)$$ and $$T=F(Y)$$, to the functor $$S\to T\to T$$. The left Kan extension along $$Y\to (X\to Y\to Z)$$ sends $$S$$ to the functor $$0\to S\to S$$, and the left Kan extension along $$(Y\to Z)\to (X\to Y\to Z)$$ sends $$S\to T$$ to $$0\to S\to T$$.
Now, suppose $$D$$ is any diagram mapping to $$\pi\circ D$$, rather than your specific example. Let the value of $$D$$ over $$X\to Y$$ be $$A\to B$$, the value over $$Y$$ alone be $$C$$ and the value over $$Y\to Z$$ be $$D\to E$$. Then to give $$D$$ as a whole is just to give functions $$C\to B$$ and $$C\to D$$. Now, $$D'$$ becomes the diagram in $$[X\to Y\to Z,\mathbf{Set}]$$ given by $$(A\to B\to B)\leftarrow (0\to C\to C)\to (0\to D\to E)$$, whose pushout is given by $$A\to B\sqcup_C D\to B\sqcup_C E$$.
The background to this possibly elaborate-looking construction is that the forgetful functor $$\mathbf{DSets}\to \mathbf{Cat}$$ is a Grothendieck opfibration via the left Kan extension functors, and such a construction always makes the total category of a Grothendieck opfibration cocomplete if the base and the fibers are, assuming the transition functors are cocontinuous.
• I see. So I was mistaken when I thought $R(Z')$ would be a two-element set. It is in fact a four-element set $\{u',v',d'',e''\}$ and the images of $d'$ and $e'$ I was looking for are simply $d''$ and $e''$, right ? It is strange that the colimit is in some way "creating" new points to accomodate the new diagram $\mathbf{C}_D$. Commented Sep 21, 2019 at 11:09