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

TL;DR

This paper develops an obstruction theory for Gr-functors between Gr-categories, linking their classification to cohomology groups and establishing conditions for their existence and equivalence.

Contribution

It introduces an obstruction element in third cohomology for Gr-functors and relates their classification to second cohomology, connecting Gr-category theory with group extension problems.

Findings

Obstruction to Gr-functor existence is in H^3(b5,c7).

Vanishing obstruction implies a bijection with H^2(b5,c7).

Every Gr-category is Gr-equivalent to a strict one.

Abstract

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.