Existence, uniqueness and functoriality of the perfect locality over a Frobenius P-category

TL;DR

This paper provides direct proofs for the existence, uniqueness, and functoriality of a certain category associated with Frobenius P-categories, advancing the understanding of their structural properties without relying on the Classification of finite simple groups.

Contribution

It offers new direct proofs for the existence and uniqueness of the perfect locality over a Frobenius P-category, and establishes its functoriality, simplifying previous complex cohomological approaches.

Findings

Proved the existence and uniqueness of the perfect locality directly.

Established the functoriality of the correspondence between categories.

Simplified the understanding of the structure of Frobenius P-categories.

Abstract

Let p be a prime, P a finite p-group and F a Frobenius P-category. The question on the existence of a suitable category Lsc extending the full subcategory of F over the set of F-selfcentralizing subgroups of P goes back to Dave Benson in 1994. In 2002 Carles Broto, Ran Levi and Bob Oliver formulate the existence and the uniqueness of the category Lsc in terms of the annulation of an obstruction 3-cohomology element and of the vanishing of a 2-cohomology group, and they state a sufficient condition for the vanishing of these n-cohomology groups. Recently, Amy Chermak has proved the existence and the uniqueness of Lsc via his objective partial groups, and Bob Oliver, following some of Chermak's methods, has also proved the vanishing of those n-cohomology groups for n > 1, both applying the Classification of the finite simple groups. Here we give direct proofs of the existence and the uniqueness of Lsc; moreover, in [11] we already show that Lsc can be completed in a suitable category L extending F and here we prove some functoriality of this correspondence.