Coproducts in Categories of q-Matroids

TL;DR

This paper explores the structure of q-matroids, introducing various maps and categories, and identifies that only the category with linear weak maps admits a coproduct, which corresponds to the direct sum of q-matroids.

Contribution

It introduces different types of maps between q-matroids and shows that only the category with linear weak maps has a coproduct, linking it to the direct sum construction.

Findings

Only the category with linear weak maps has a coproduct.

The coproduct in this category is the direct sum of q-matroids.

Various types of maps between q-matroids are characterized and categorized.

Abstract

q-Matroids form the q-analogue of classical matroids. In this paper we introduce various types of maps between q-matroids. These maps are not necessarily linear, but they map subspaces to subspaces and respect the q-matroid structure in certain ways. The various types of maps give rise to different categories of q-matroids. We show that only one of these categories possesses a coproduct. This is the category where the morphisms are linear weak maps, that is, the rank of the image of any subspace is not larger than the rank of the subspace itself. The coproduct in this category is the very recently introduced direct sum of q-matroids.