An $\mathcal{O}$-monoidal Grothendieck construction
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| Format: | Preprint |
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2024
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| _version_ | 1866911821528563712 |
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| author | Haderi, Redi Stern, Walker H. |
| author_facet | Haderi, Redi Stern, Walker H. |
| contents | Given an operad $\mathcal{O}$, we define a notion of weak $\mathcal{O}$-monoids -- which we term $\mathcal{O}$-pseudomonoids -- in a 2-category. In the special case with the 2-category in question is the 2-category $\mathsf{Cat}$ of categories, this yields a notion of $\mathcal{O}$-monoidal category, which in the case of the associative and commutative operads retrieves unbiased notions of monoidal and symmetric monoidal categories, respectively. We carefully unpack the definition of $\mathcal{O}$-monoids in the 2-categories of discrete fibrations and of category-indexed sets. Using the classical Grothendieck construction, we thereby obtain an $\mathcal{O}$-monoidal Grothendieck construction relating lax $\mathcal{O}$-monoidal functors into Set to strict $\mathcal{O}$-monoidal functors which are also discrete fibrations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_01031 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An $\mathcal{O}$-monoidal Grothendieck construction Haderi, Redi Stern, Walker H. Category Theory 18N10 (primary), 18M60 (secondary) Given an operad $\mathcal{O}$, we define a notion of weak $\mathcal{O}$-monoids -- which we term $\mathcal{O}$-pseudomonoids -- in a 2-category. In the special case with the 2-category in question is the 2-category $\mathsf{Cat}$ of categories, this yields a notion of $\mathcal{O}$-monoidal category, which in the case of the associative and commutative operads retrieves unbiased notions of monoidal and symmetric monoidal categories, respectively. We carefully unpack the definition of $\mathcal{O}$-monoids in the 2-categories of discrete fibrations and of category-indexed sets. Using the classical Grothendieck construction, we thereby obtain an $\mathcal{O}$-monoidal Grothendieck construction relating lax $\mathcal{O}$-monoidal functors into Set to strict $\mathcal{O}$-monoidal functors which are also discrete fibrations. |
| title | An $\mathcal{O}$-monoidal Grothendieck construction |
| topic | Category Theory 18N10 (primary), 18M60 (secondary) |
| url | https://arxiv.org/abs/2404.01031 |