Cohomological classification of Ann-functors

Regular Ann-functor classification problem has been solved with Shukla cohomology. In this paper, we would like to present a solution to the above problem in the general case and in the case of strong Ann-functors with, respectively, Mac Lane cohomology and Hochschild cohomology.

💡 Research Summary

The paper addresses the classification problem for Ann‑functors, which are morphisms between Ann‑categories—categorical structures equipped with two interacting binary operations, addition (⊕) and multiplication (⊗), together with coherence data expressing associativity, commutativity, and distributivity. Earlier work solved the classification of regular Ann‑functors (those whose unit morphism is the identity) by using Shukla cohomology H³ₛ. However, many natural situations involve either non‑regular Ann‑functors, where the unit morphism need not be the identity, or strong Ann‑functors, which preserve units up to isomorphism and satisfy additional strictness conditions. The present work extends the cohomological classification to these broader contexts by employing two different cohomology theories: Mac Lane cohomology for the general case and Hochschild cohomology for the strong case.

The authors begin by recalling the definition of an Ann‑category and the data required for an Ann‑functor: a pair of functors preserving ⊕ and ⊗, together with natural transformations φ and ψ that encode the distributive law and unit compatibility. They package this data as a triple (F, φ, ψ). By translating the coherence axioms into equations on a 3‑cochain in the Mac Lane complex C³ₘₗ(R, M) (where R is the underlying ring and M the bimodule of morphisms), they show that the obstruction to the existence of a genuine Ann‑functor is precisely the cohomology class