# Residually finite categories

## TL;DR

This paper introduces residual finiteness for categories, demonstrating that certain categories like free categories and subcategories of vector spaces possess this property, which implies several algebraic and computational advantages.

## Contribution

It extends the concept of residual finiteness from groups to categories and establishes key properties and examples of residually finite categories.

## Key findings

- Free categories are residually finite.
- Finitely generated subcategories of vector spaces are residually finite.
- Residually finite categories are Hopfian and have solvable word problem.

## Abstract

We introduce the notion of residual finiteness for categories. In analogy with the group-theoretic setting, we prove that free categories and finitely generated subcategories of finite-dimensional vector spaces are residually finite. Moreover, finitely generated residually finite categories are Hopfian and finitely presented residually finite categories have solvable word problem.