Stacks of Ann-Categories and their morphisms

TL;DR

This paper demonstrates that ann-categories can be represented by crossed bimodules and that morphisms between them correspond to specific spans called bimodule butterflies, establishing an equivalence of their morphism groupoids.

Contribution

It introduces a presentation of ann-categories via crossed bimodules and characterizes their morphisms through bimodule butterflies, linking categorical and algebraic structures.

Findings

Ann-categories admit presentation by crossed bimodules

Morphisms between ann-categories are equivalent to bimodule butterflies

Groupoid of morphisms is characterized by spans called butterflies

Abstract

We show that $\mathit{ann}$-categories admit a presentation by crossed bimodules, and prove that morphisms between them can be expressed by special kinds spans between the presentations. More precisely, we prove the groupoid of morphisms between two $\mathit{ann}$-categories is equivalent to that of bimodule butterflies between the presentations. A bimodule butterfly is a specialization of a butterfly, i.e. a special kind of span or fraction, between the underlying complexes