Schreier Theory of Track Categories

TL;DR

This paper extends non-abelian cohomology theory to classify non-abelian track categories, generalizing previous abelian results and providing a framework for understanding complex categorical structures.

Contribution

It generalizes the classification of abelian track categories to non-abelian cases using non-abelian cohomology, specifically for categories with certain functor and natural system conditions.

Findings

Classifies non-abelian track categories with given underlying and homotopy categories.

Establishes a correspondence between cohomology classes and track categories.

Extends previous abelian cohomology results to non-abelian settings.

Abstract

This paper is a continuation of our study of non-abelian Baues-Wirsching cohomologies. In our previous paper, we defined second non-abelian cohomology H2(C;D) of a small category C with coefficients in a so-called centralised natural system D. We proved that H2(C;D) classifies linear extensions of C by D, generalising the corresponding result for abelian natural systems. For an abelian natural system D, the third cohomology classifies certain abelian track categories. A track category is a 2-category where all 2-morphisms are isomorphisms. A track category is called abelian if for every 1-morphism f, the group Aut(f) is abelian. In a similar fashion to the above, we want to generalise this result for non-abelian track categories. In this paper we solve this problem for the following important case: Given categories K and C and a functor ? p: K -> C, which is identity on objects and surjective on morphisms, and G, a centralised natural system of groups on K, we describe the equivalence classes of all track categories T for which K is the underlying category and C is the homotopy category and G_f = Aut(f).