Invertibility in Category Representations

TL;DR

This paper characterizes when representations of small categories can be factored through inverse categories, providing a numerical, decidable condition for finite categories, with implications for topological data analysis.

Contribution

It introduces a numerical criterion to determine when a category representation factors through an inverse category, enhancing understanding of invertibility in category representations.

Findings

Provides a decidable numerical condition for finite categories

Characterizes factorization through inverse categories

Applies to persistence modules in topological data analysis

Abstract

Inverse categories are categories in which every morphism x has a unique pseudo-inverse y in the sense that xyx=x and yxy=y. Persistence modules from topological data analysis and similarly decomposable category representations factor through inverse categories. This paper gives a numerical condition, decidable when the indexing category is finite, characterizing when a representation of a small category factors through an inverse category.