---
date: 2021-05-22
tags: [ web/quick-notes ]
---


# 2021-05-22

## 23:27

Tags:
#personal/idle-thoughts #higher-algebra/category-theory #higher-algebra/infty-cats 

### 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](infinity%20categories.md).

### 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?