A symmetric monoidal Comparison Lemma

Josefien Kuijper

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arXiv · 2309.11444

A symmetric monoidal Comparison Lemma

Josefien Kuijper

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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.

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Josefien Kuijper. 2023-09-20. A symmetric monoidal Comparison Lemma. https://arxiv.org/abs/2309.11444

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