# Noteworthy examples of finite categories

So far all the finite categories I have encountered fall into one of these c̶a̶t̶e̶g̶o̶r̶i̶e̶s̶ sets:

• finite monoids
• finite preorders
• just formal devices to explain, what a "diagram" in another (infinite) category is

Are there any other finite categories, which are not monoids or preorders, which are interesting by themselves?

• To the extent that we really mainly care about monoids because they act on things, arguably your first example is a special case of your third example. – Eric Wofsey Jul 1 '16 at 5:38

One example is what is called a fusion system.

A fusion system is a category where the objects are the subgroups of some fixed $p$-group $S$ and where the morphisms is a subset of the set of injective homomorphisms between the subgroups which contains all those induced by conjugation by some element from $S$.
Further, it is required that any morphism $\varphi$ from $P$ to $Q$ factors through the inclusion of $\varphi(P)$ into $Q$ and that the inverse homomorphism $\varphi^{-1}: \varphi(P)\to P$ is also in the category.

These are meant as a generalization of the fusion structure on the set of subgroups of a $p$-Sylow subgroup of a finite group, and they have been studied extensively over the past 10 years or so, especially by people in various areas if algebraic topology.

• This reminds me of the orbit category of a finite group, the domain of a Mackey functor, which has quite a lot of structure. – Kevin Arlin Jul 1 '16 at 16:55

I like the Mathieu Groupoid $M_{13}$.

Also, for any group $G$ acting on a set $X$ there is the action groupoid $X/\!/G$ with objects the elements of $X$ and with morphisms $x_1 \to x_2$ given by the elements of $G$ such that $g \cdot x_1 =x_2$. (Of course, if $G$ and $X$ are finite, so is $X/\!/G$.)

(Maybe these don't quite count, as any groupoid is equivalent to a disjoint union of groups and you already mentioned monoids...)