An Enriched Approach to the Strictification of $(\infty,1)$-Categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909825669005312 |
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| author | Strong, Kimball |
| author_facet | Strong, Kimball |
| contents | We define a functor which takes in an $(\infty,1)$-category and outputs an $(ω,1)$-category, the natural maximally "strict" version of an $(\infty,1)$-category. We do this by modeling $(\infty,1)$-categories as categories enriched in $\infty$-groupoids, and then "locally strictifying" (applying the strictification of $\infty$-groupoids to each hom space) to obtain a category enriched in $ω$-groupoids with respect to the Gray tensor product, followed by "globally strictifying" (strictifying the enrichment from the Gray tensor product to the cartesian product) to obtain a category cartesian-enriched in $ω$-groupoids, which is equivalently an $(ω,1)$-category. We conjecture that this functor is conservative, and prove this for two dual special cases: $2$-truncated and $2$-connected $(\infty,1)$-categories. Along the way, we construct a sort of "incoherent walking $(ω,1)$-equivalence," which gives a simpler description of the coherent path lifting condition for fibrations of $(ω,1)$-categories, only involving cells of dimension $\le 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04254 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Enriched Approach to the Strictification of $(\infty,1)$-Categories Strong, Kimball Category Theory Algebraic Topology 18N30 (Primary) 18N60 (Secondary) We define a functor which takes in an $(\infty,1)$-category and outputs an $(ω,1)$-category, the natural maximally "strict" version of an $(\infty,1)$-category. We do this by modeling $(\infty,1)$-categories as categories enriched in $\infty$-groupoids, and then "locally strictifying" (applying the strictification of $\infty$-groupoids to each hom space) to obtain a category enriched in $ω$-groupoids with respect to the Gray tensor product, followed by "globally strictifying" (strictifying the enrichment from the Gray tensor product to the cartesian product) to obtain a category cartesian-enriched in $ω$-groupoids, which is equivalently an $(ω,1)$-category. We conjecture that this functor is conservative, and prove this for two dual special cases: $2$-truncated and $2$-connected $(\infty,1)$-categories. Along the way, we construct a sort of "incoherent walking $(ω,1)$-equivalence," which gives a simpler description of the coherent path lifting condition for fibrations of $(ω,1)$-categories, only involving cells of dimension $\le 3$. |
| title | An Enriched Approach to the Strictification of $(\infty,1)$-Categories |
| topic | Category Theory Algebraic Topology 18N30 (Primary) 18N60 (Secondary) |
| url | https://arxiv.org/abs/2510.04254 |