Generalized multicategories: change-of-base, embedding, and descent
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arXiv
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| Natura: | Preprint |
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2023
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| author | Prezado, Rui Nunes, Fernando Lucatelli |
| author_facet | Prezado, Rui Nunes, Fernando Lucatelli |
| contents | Via the adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Span}(\mathcal V) \to \mathcal V \text{-} \mathsf{Mat} $ and a cartesian monad $ T $ on an extensive category $ \mathcal V $ with finite limits, we construct an adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Cat}(T,\mathcal V) \to (\overline T, \mathcal V)\text{-}\mathsf{Cat} $ between categories of generalized enriched multicategories and generalized internal multicategories, provided the monad $ T $ satisfies a suitable condition, which is satisfied by several examples.
We verify, moreover, that the left adjoint is fully faithful, and preserves pullbacks, provided that the copower functor $ - \boldsymbol{\cdot} 1 \colon \mathsf{Set} \to \mathcal V $ is fully faithful. We also apply this result to study descent theory of generalized enriched multicategorical structures.
These results are built upon the study of base-change for generalized multicategories, which, in turn, was carried out in the context of categories of horizontal lax algebras arising out of a monad in a suitable 2-category of pseudodouble categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08084 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalized multicategories: change-of-base, embedding, and descent Prezado, Rui Nunes, Fernando Lucatelli Category Theory 18B10, 18B15, 18B50, 18D65, 18N10, 18N15 Via the adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Span}(\mathcal V) \to \mathcal V \text{-} \mathsf{Mat} $ and a cartesian monad $ T $ on an extensive category $ \mathcal V $ with finite limits, we construct an adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Cat}(T,\mathcal V) \to (\overline T, \mathcal V)\text{-}\mathsf{Cat} $ between categories of generalized enriched multicategories and generalized internal multicategories, provided the monad $ T $ satisfies a suitable condition, which is satisfied by several examples. We verify, moreover, that the left adjoint is fully faithful, and preserves pullbacks, provided that the copower functor $ - \boldsymbol{\cdot} 1 \colon \mathsf{Set} \to \mathcal V $ is fully faithful. We also apply this result to study descent theory of generalized enriched multicategorical structures. These results are built upon the study of base-change for generalized multicategories, which, in turn, was carried out in the context of categories of horizontal lax algebras arising out of a monad in a suitable 2-category of pseudodouble categories. |
| title | Generalized multicategories: change-of-base, embedding, and descent |
| topic | Category Theory 18B10, 18B15, 18B50, 18D65, 18N10, 18N15 |
| url | https://arxiv.org/abs/2309.08084 |