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!
Best Answer
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.