Extending homotopy theories across adjunctions

TL;DR

This paper explores how to extend homotopy theories across different types of monoidal functors, enabling broader equivalences between categories with weaker structure-preserving maps.

Contribution

It establishes conditions under which weak equivalences can be transported across adjunctions, broadening the scope of homotopy theory applications.

Findings

Transport of weak equivalences under mild conditions

Equivalence of homotopy theories for various algebraic structures

Applicability to symmetric monoidal and diagram categories

Abstract

Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $\Gamma$-categories.