The category of matroids

TL;DR

This paper explores the categorical structure of matroids, revealing their limits, functors, adjunctions, and the functoriality of certain combinatorial constructions, advancing the theoretical understanding of matroid theory.

Contribution

It provides a detailed analysis of the category of matroids, identifying key categorical properties, functors, and adjunctions, and clarifies which matroid constructions are functorial.

Findings

The category of matroids has coproducts and equalizers but lacks products and coequalizers.

There are faithful functors from graphs and vector spaces to matroids.

A functor to geometric lattices is nearly full, and various adjunctions and free constructions are identified.

Abstract

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial.