On Gr-Functors between Gr-Categories: Obstruction theory for Gr-Functors of the type $(varphi,f)$

Each Gr-functor of the type $(\varphi,f)$ of a Gr-category of the type $(\Pi,\C)$ has the obstruction be an element $\overline{k}\in H^3(\Pi,\C).$ When this obstruction vanishes, there exists a bijection between congruence classes of Gr-functors of the type $(\varphi,f)$ and the cohomology group $H^2(\Pi,\C).$ Then the relation of Gr-category theory and the group extension problem can be established and used to prove that each Gr-category is Gr-equivalent to a strict one.

💡 Research Summary

The paper develops a complete obstruction‑theoretic framework for Gr‑functors between Gr‑categories, focusing on those of the “type $(\varphi,f)$”. A Gr‑category is presented as a pair $(\Pi,\mathcal C)$ where $\Pi$ is a group of objects and $\mathcal C$ is a $\Pi$‑module of automorphisms; the associativity defect of the monoidal structure is encoded by a normalized 3‑cocycle $k\in Z^{3}(\Pi,\mathcal C)$.

A Gr‑functor of type $(\varphi,f)$ consists of a group homomorphism $\varphi:\Pi\to\Pi’$ on objects and a $\Pi$‑module homomorphism $f:\mathcal C\to\mathcal C’$ on morphisms. The paper shows that such a pair can be lifted to an actual monoidal functor precisely when a certain cohomology class, called the obstruction, vanishes. This obstruction is the element
\