# 2021-05-22

## 23:27

### Cats with cats of morphisms

Terrible attempt at a way around ZFC: axiomatically define a category the way one axiomatizes Euclidean geometry:

• A collection of points (objects)
• For every two points, a category of morphisms
• Plus the usual composition axiom

This makes the definition infinitely recursive, which might be a problem. One could truncate this by asking the 1st iteration of taking “the hom category” to result in a discrete category: some objects but no morphisms between distinct objects. This is definitely taken care of by infinity categories.

### Set as a category freely generated under colimits?

Define the category of sets by specifying a single point as an initial object, then freely taking powersets and unions. I think you at least get something whose nerve is the same as the nerve of the category of finite sets. I think one can also realize these operations at the categorical level: powersets are like exponentials $$2^X$$, you can get disjoint unions from limits, and maybe usual unions/intersections from pushouts/pullbacks?