Explicitly describe colimits in $\mathsf{Set}$

Question

I just started learning category theory a couple months ago. In my understanding, there is a nice fact about the category of sets that one can explicitly describe limits. If $F:J \to \mathsf{Set}$ is a diagram, then $\lim F$ is just the set $\mathscr{S}$ of all cones on $F$ over the singleton set $\{*\}$ (provided that the cones form a set). This is because a cone over a set $S$ is nothing other than a cone over each $s\in S$, since elements in a set have no relations. Therefore there exists a unique morphism of cones $S \to \mathscr{S}$.

I've been trying to formulate a dual statement. Is there a convenient way to describe the colimit of a diagram in $\mathsf{Set}$?

Answer 1

It is not as easy to see, but the following is true:

Let $X : \mathcal{J} \to \mathbf{Set}$ be a diagram and let $\mathbf{El} (X)$ be the category defined below:

• The objects are pairs $(j, x)$ where $j$ is an object in $\mathcal{J}$ and $x$ is an element of $X j$.
• The morphisms $(j, x) \to (k, y)$ are morphisms $f : j \to k$ in $\mathcal{J}$ such that $(X f) (x) = y$.
• Composition and identities are inherited from $\mathcal{J}$.

Then $\varinjlim_\mathcal{J} X$ can be identified with the set of connected components of $\mathbf{El} (X)$. (Two objects in a category if there is a zigzag of arrows connecting them.)