Note on the universality and the functoriality of the perfect F-locality

TL;DR

This paper extends the universality and functoriality properties of localizing functors to perfect localities in Frobenius P-categories, removing previous restrictions and strengthening the theoretical framework.

Contribution

It generalizes the functoriality of perfect localities by removing the Abelian hypothesis and moving from localizing functors to perfect localities.

Findings

Established functoriality for perfect localities in the strongest form

Removed the Abelian group restriction in the target

Extended universality results to a broader class of localities

Abstract

In "Frobenius Categories versus Brauer Blocks" we have proved some universality of the so-called localizing functor associated with a Frobenius $P$-category $F$, where $P$ is a finite $p$-group, with respect to the coherent $F$-localities $(\tau,L,\pi)$ such that the contravariant functor $Ker (\pi)$ maps any subgroup of $P$ to an Abelian $p$-group. The purpose of this Note is both to move from the localizing functor to the perfect locality associated with $F$ and to remove the Abelian hypothesis in the target. As a consequence, we get the functoriality for the perfect localities in the strongest form, improving the Theorem 9.15 in "Existence, uniqueness and functoriality of the perfect locality over a Frobenius $P$-category".