Lax monoidal adjunctions, two-variable fibrations and the calculus of mates

Rune Haugseng; Fabian Hebestreit; Sil Linskens; Joost Nuiten

doi:10.1112/plms.12548

Lax monoidal adjunctions, two-variable fibrations and the calculus of mates

Rune Haugseng^*, Fabian Hebestreit, Sil Linskens, Joost Nuiten

^*Corresponding author for this work

Mathematical Science

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

We provide a calculus of mates for functors to the (Formula presented.) -category of (Formula presented.) -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 (Formula presented.) -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 (Formula presented.) -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 (Formula presented.) -categories.

Original language	English
Pages (from-to)	889-957
Number of pages	69
Journal	Proceedings of the London Mathematical Society
Volume	127
Issue number	4
Early online date	24 Aug 2023
DOIs	https://doi.org/10.1112/plms.12548
Publication status	Published - Oct 2023

Bibliographical note

Funding Information:
During the preparation of this manuscript FH and SL were members of the Hausdorff Center for Mathematics at the University of Bonn funded by the German Research Foundation (DFG), grant no. EXC 2047. FH and JN were further supported by the European Research Council (ERC) through the grants ‘Moduli spaces, Manifolds and Arithmetic’, grant no. 682922, and ‘Derived Symplectic Geometry and Applications’, grant no. 768679, respectively.

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© 2023 The Authors. Proceedings of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
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Cite this

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abstract = "We provide a calculus of mates for functors to the (Formula presented.) -category of (Formula presented.) -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 (Formula presented.) -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 (Formula presented.) -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 (Formula presented.) -categories.",

author = "Rune Haugseng and Fabian Hebestreit and Sil Linskens and Joost Nuiten",

note = "Funding Information: During the preparation of this manuscript FH and SL were members of the Hausdorff Center for Mathematics at the University of Bonn funded by the German Research Foundation (DFG), grant no. EXC 2047. FH and JN were further supported by the European Research Council (ERC) through the grants {\textquoteleft}Moduli spaces, Manifolds and Arithmetic{\textquoteright}, grant no. 682922, and {\textquoteleft}Derived Symplectic Geometry and Applications{\textquoteright}, grant no. 768679, respectively. ",

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Lax monoidal adjunctions, two-variable fibrations and the calculus of mates

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