Familiar category that is a colimit over a diagram of finite categories

Question

Can anyone give an example of a "familiar" category that is a colimit over a diagram of finite categories? I am not sure how to define familiar, but maybe SET is too familiar and if only one person has ever mentioned it, maybe that's not familiar enough.

Answer 1

Every category $\mathcal{C}$ is the colimit of the following (possibly large) diagram of finite categories. For each object, each morphism, and each pair of composable morphisms of $\mathcal{C}$, we have an object in the diagram. For an object $A$ in $\mathcal{C}$, the finite category in our diagram is $1$, the category with one object and one morphism, which maps to $\mathcal{C}$ by sending the one object to $A$. For a morphism $f:A\to B$ of $\mathcal{C}$, the finite category in our diagram is $2$, the category with two objects $X$ and $Y$ and no non-identity morphisms except one morphism $g:X\to Y$, which maps to $\mathcal{C}$ by sending $g$ to $f$. For a composable pair of morphisms $A\to B\to C$ of $\mathcal{C}$, the finite category in our diagram is $3$, the category with three objects $X$, $Y$, and $Z$ and single non-identity morphisms $X\to Y$ and $Y\to Z$ and their composition $X\to Z$. This of course maps to $\mathcal{C}$ by sending $X\to Y\to Z$ to $A\to B\to C$ in the obvious way.

The morphisms in our diagram are then all functors between these copies of $1$, $2$, and $3$ which commute with their maps to $\mathcal{C}$. For instance, that means that we have functors from each copy of $1$ to all the copies of $2$ and $3$ that involve that same object of $\mathcal{C}$, and we have functors from each copy of $2$ to all the copies of $3$ that involve that same morphism of $\mathcal{C}$. In this way, the morphisms in our diagram tell us how to "glue together" these copies of $1$, $2$, and $3$ to get $\mathcal{C}$.

The fact that $\mathcal{C}$ is the colimit of this diagram is then pretty much automatic. A functor on $\mathcal{C}$ is determined by what it does on objects and morphisms (i.e., what it does when composed with each of the functors $1\to\mathcal{C}$ and $2\to\mathcal{C}$ from our diagram), with the restriction that it must preserve identities and composition (which follows from compatibility with the morphisms of the form $2\to 1$ and $2\to 3$ in our diagram).