Partial actions of inverse categories and their algebras

TL;DR

This paper introduces and studies partial and global actions of inverse categories on posets, explores their algebraic structures, and establishes Morita equivalences between their convolution algebras, extending concepts from inverse semigroup theory.

Contribution

It develops the theory of partial and global actions of inverse categories, introduces Bernoulli actions and Szendrei expansions, and extends enlargement and Morita equivalence concepts to inverse categories.

Findings

Bernoulli actions lead to Szendrei expansions of inverse categories

Enlargements of inverse categories induce Morita equivalent convolution algebras

Cauchy completions of enlarged categories are equivalent

Abstract

In this work we introduce partial and global actions of inverse categories on posets in two variants, fibred actions and actions by symmetries. We study in detail actions of an inverse category $\mathcal{C}$ on specific subposets of the poset of finite subsets of $\mathcal{C}$, the Bernoulli actions. We show that to each fibred action of an inverse category on a poset there corresponds another inverse category, the semidirect product associated to the action. The Bernoulli actions give rise to the Szendrei expansions of $\mathcal{C}$, which define a endofunctor of the category of inverse categories. We extend the concept of enlargement from inverse semigroup theory to, and we show that if $\mathcal{D}$ is an enlargement of $\mathcal{C}$ then their Cauchy completions are equivalent categories; in particular, some pairs corresponding to partial and global Bernoulli actions are enlargements. We conclude by studying convolution algebras of finite inverse categories and showing that if $\mathcal{D}$ is an enlargement of $\mathcal{C}$ then their convolution algebras are Morita equivalent. Furthermore, using Kan extensions we also analyze the infinite case.