A Hochschild Cohomology Comparison Theorem for prestacks

We generalize and clarify Gerstenhaber and Schack’s “Special Cohomology Comparison Theorem”. More specifically we obtain a fully faithful functor between the derived categories of bimodules over a prestack over a small category U and the derived category of bimodules over its corresponding fibered category. In contrast to Gerstenhaber and Schack we do not have to assume that U is a poset.

💡 Research Summary

The paper presents a substantial generalization of the Gerstenhaber‑Schack Special Cohomology Comparison Theorem (SCT) by removing the restrictive hypothesis that the indexing category U be a poset. The authors work with a prestack 𝔄 over an arbitrary small category U, where a prestack consists of an algebra 𝔄_U for each object U and restriction morphisms ρ_f for each arrow f:U→V, together with coherent 2‑cell data η_{f,g}:ρ_f∘ρ_g⇒ρ_{gf} that encode higher compatibility. This enriched structure allows one to treat 𝔄 as a genuine 2‑category object rather than a mere diagram of algebras.

The central construction is the passage from the prestack 𝔄 to its associated fibered category 𝔉(𝔄). Objects of 𝔉(𝔄) are pairs (U, A) with A∈𝔄_U, and morphisms are given by a base arrow f:U→V together with a morphism α: A→ρ_f(B) in the fiber over U. This fibered category captures both the horizontal (base) and vertical (algebraic) directions of the prestack in a single categorical object.

On both sides the authors consider the derived category of bimodules. For the prestack they define Mod‑Bimod(𝔄) as the category of complexes equipped with compatible left and right 𝔄‑module structures, together with differentials that respect the restriction maps and the 2‑cell data. Analogously, Mod‑Bimod(𝔉(𝔄)) consists of bimodule complexes over the fibered category, where the action of a morphism (f,α) combines the base change along f with the algebraic action via α.

The main theorem asserts the existence of a fully faithful exact functor \