What are a couple of examples of finite sized but interesting categories?

Question

I'm studying category theory and, given that I don't have a background in topology, I'm struggling to think of some finite categories that interesting.

The main one I know of is finite preorders -- I find this category interesting because it has products, sums, etc. I'd love a couple of other categories like that that I could use to better understand (by graphing out the objects and morphisms) things like equalizers, pullbacks, etc.

I could of course do this by construction, but I'd much prefer some finite categories that include these ideas in a useful way.

Thank you!

Answer 1

One useful type of finite categories are fusion systems. The motivating example for this notion is the following: Take a finite group $G$ and fix a sylow-$p$-subgroup $S\leq G$. Define a category $\mathcal{F}$ as follows. Objects are all subgroups of $S$ and morphisms $P\to Q$ are given by all the conjugation maps $x\mapsto gxg^{-1}$ that take $P$ into $Q$. This category captures a lot of the $p$-modular representation theory of $G$.

A fusion system is a generalization of this kind of structure. It is still a category whose objects are subgroups of some $p$-group $S$ and whose morphisms are injective group homomorphisms between them satisfying some list of axioms that is modelled on the example above. There are exotic fusion systems that satisfy these axioms but do not come from any group. This is one of the bigger complications that can arise in modular representation theory.

Answer 2

Taking the attitude that a category is interesting not so much in its own right as for the functors into or out of it, I like the category with just two objects, called $A$ and $V$, and with just four morphisms, namely the two identity morphisms and two other morphisms, called $s$ and $t$, both pointing from $A$ to $V$. A functor $F$ from this category to the category of sets is "the same thing" as a directed graph (allowing loops and multiple edges). $F(V)$ is the set of vertices of the graph, $F(A)$ is the set of arrows, and $F(s)$ and $F(t)$ assign to each arrow its source and its target, respectively.

Continuing to take the same attitude, I also like any finite group as an example of a category with just one object; the morphisms are the elements of the group and composition is the group's operation. A functor from this to the category of sets is a permutation action of the group. A functor from the group to the category of vector spaces over a field $k$ is a $k$-linear representation of the group.

Answer 3

A well-known example is the equivalence between groups and and one-object categories with all morphisms invertible (i.e. all morphisms are isomorphisms). The group is finite iff the category is.

A more exotic example of a finite category with all morphisms invertible is Conway's $M_{13}$ groupoid (see also this blog entry by John Baez, who is fond of most things categorical). Conway calls it "$M_{13}$" because there are $13$ objects each of whose automorphism groups is isomorphic with the Mathieu group $M_{12}$ (and any two objects are connected by $|M_{12}|$ isomorphisms).