# Sizes and filtrations in accessible categories

## TL;DR

This paper explores the concept of internal size in accessible categories, establishing set-theoretic properties and filtrations, including L"owenheim-Skolem theorems, under certain set-theoretic assumptions.

## Contribution

It generalizes previous results by proving new set-theoretic properties and filtrations for accessible categories using internal sizes, extending the theory significantly.

## Key findings

- Large accessible categories contain objects of all high enough internal sizes.
- Accessible categories with directed colimits have filtrations of objects by smaller objects.
- Under the singular cardinal hypothesis, certain size-related objects always exist.

## Abstract

Accessible categories admit a purely category-theoretic replacement for cardinality: the internal size. Generalizing results and methods from arXiv:1708.06782, we examine set-theoretic problems related to internal sizes and prove several L\"owenheim-Skolem theorems for accessible categories. For example, assuming the singular cardinal hypothesis, we show that a large accessible category has an object in all internal sizes of high-enough cofinality. We also prove that accessible categories with directed colimits have filtrations: any object of sufficiently high internal size is (the retract of) a colimit of a chain of strictly smaller objects.