Homotopy coherent centers versus centers of homotopy categories

TL;DR

This paper introduces homotopy coherent centers as an extension of classical centers to categories with homotopy theory, providing new invariants and tools like obstruction theory and spectral sequences for their analysis.

Contribution

It develops the concept of homotopy coherent centers, connecting them to Hochschild cohomology, and offers methods for computing and realizing elements within these centers.

Findings

Homotopy coherent centers generalize classical centers in homotopical contexts.

A spectral sequence is constructed to compute homotopy groups of these centers.

Obstruction theory is developed for realizing elements in homotopy category centers.

Abstract

Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are not necessarily linear or stable, and we argue that they are more appropriate to higher categorical contexts than the centers of their homotopy or derived categories. Among many other things, we present an obstruction theory for realizing elements in the centers of homotopy categories, and a Bousfield-Kan type spectral sequence that computes the homotopy groups. Nontrivial classes of examples are given as illustration throughout.