Lax monoidal adjunctions, two-variable fibrations and the calculus of mates
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866911824659611648 |
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| author | Haugseng, Rune Hebestreit, Fabian Linskens, Sil Nuiten, Joost |
| author_facet | Haugseng, Rune Hebestreit, Fabian Linskens, Sil Nuiten, Joost |
| contents | We provide a calculus of mates for functors to the $\infty$-category of $\infty$-categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do not necessarily preserve (co)cartesian edges. As a sample application we obtain an equivalence between lax symmetric monoidal structures on right adjoint functors and oplax symmetric monoidal structures on the left adjoint functors between symmetric monoidal $\infty$-categories that is compatible with both horizontal and vertical composition of such structures.
As the technical heart of the paper we study various new types of fibrations over a product of two $\infty$-categories. In particular, we show how they can be dualised over one of the two factors and how they encode functors out of the Gray tensor product of $(\infty, 2)$-categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_08808 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates Haugseng, Rune Hebestreit, Fabian Linskens, Sil Nuiten, Joost Category Theory Algebraic Topology We provide a calculus of mates for functors to the $\infty$-category of $\infty$-categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do not necessarily preserve (co)cartesian edges. As a sample application we obtain an equivalence between lax symmetric monoidal structures on right adjoint functors and oplax symmetric monoidal structures on the left adjoint functors between symmetric monoidal $\infty$-categories that is compatible with both horizontal and vertical composition of such structures. As the technical heart of the paper we study various new types of fibrations over a product of two $\infty$-categories. In particular, we show how they can be dualised over one of the two factors and how they encode functors out of the Gray tensor product of $(\infty, 2)$-categories. |
| title | Lax monoidal adjunctions, two-variable fibrations and the calculus of mates |
| topic | Category Theory Algebraic Topology |
| url | https://arxiv.org/abs/2011.08808 |