A symmetric monoidal Comparison Lemma
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915048355528704 |
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| author | Kuijper, Josefien |
| author_facet | Kuijper, Josefien |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_11444 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A symmetric monoidal Comparison Lemma Kuijper, Josefien Category Theory 18F20 (primary) 14F06, 18F20 (secondary) 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. |
| title | A symmetric monoidal Comparison Lemma |
| topic | Category Theory 18F20 (primary) 14F06, 18F20 (secondary) |
| url | https://arxiv.org/abs/2309.11444 |