A symmetric monoidal Comparison Lemma

TL;DR

This paper establishes a symmetric monoidal version of the Comparison Lemma, showing equivalences between lax and strong symmetric monoidal hypersheaves under certain conditions, with applications to cohomology theories.

Contribution

It extends the Comparison Lemma to symmetric monoidal contexts, providing new equivalences between lax and strong hypersheaves in higher category theory.

Findings

Proves a symmetric monoidal Comparison Lemma for hypersheaves.

Shows equivalence between lax and strong symmetric monoidal hypersheaves under certain conditions.

Applies the result to hypersheaves encoding cohomology theories.

Abstract

In this note we study symmetric monoidal functors from a symmetric monoidal 1-category to a cartesian symmetric monoidal $\infty$-category, which are in addition hypersheaves for a certain topology. We prove a symmetric monoidal version of the Comparison Lemma, for lax as well as strong symmetric monoidal hypersheaves. For a strong symmetric monoidal functor between symmetric monoidal 1-categories with topologies generated by suitable cd-structures, we show that if the conditions of the Comparison Lemma are satisfied, then there is also an equivalence between categories of lax and strong symmetric monoidal hypersheaves respectively, taking values in a complete cartesian symmetric monoidal $\infty$-category. As an application of this result, we prove a lax symmetric monoidal version of our previous result about hypersheaves that encode compactly supported cohomology theories.