Categories as models on a suitable algebraic theory

TL;DR

This paper explores how categories and groupoids serve as models for a specific algebraic theory based on graphs, demonstrating that finitely presentable models correspond to finitely presentable objects within this framework.

Contribution

It introduces a novel perspective on categories and groupoids as models for Lawvere ${rak Gr}$-theories, establishing a link between finite presentability of models and objects.

Findings

Categories and groupoids can be modeled as Lawvere ${rak Gr}$-theories.

Finitely presentable models are equivalent to finitely presentable objects.

Provides a new algebraic perspective on categorical structures.

Abstract

We explain how categories, and groupoids, can be seen as models for a Lawvere ${\mathfrak Gr}$-theory, where ${\mathfrak Gr}$ is the category of graphs, and show that for Lawvere ${\mathfrak Gr}$-theories finitely presentable models are finitely presentable objects.