On a category of cluster algebras

TL;DR

This paper introduces a new categorical framework for cluster algebras with fixed initial seeds, exploring its structural properties and connections to surface geometry.

Contribution

It defines a category of cluster algebras with fixed seeds, characterizes morphisms, and links the category to surface geometry in the context of cluster algebras from surfaces.

Findings

The category has countable coproducts but no products.

Isomorphisms and monomorphisms are characterized within the category.

Connections between the category and surface geometry are established.

Abstract

We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and provide combinatorial methods for constructing special classes of monomorphisms and epimorphisms. In the case of cluster algebras from surfaces, we describe interactions between this category and the geometry of the surfaces.