On Quasi-Categories of Comodules and Landweber Exactness

Torii, Tashiyuki

doi:10.1007/978-981-15-1588-0_11

Cited by 7 publications

(7 citation statements)

References 31 publications

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“…In particular, he has proved that the right adjoint of a strong monoidal functor is lax monoidal between monoidal ∞-categories in [5]. In [6] we showed that the right adjoint of an oplax monoidal functor is lax monoidal. Haugseng [2] and Hebestreit-Linskens-Nuiten [3] independently proved that the ∞-category Mon…”

Section: Introductionmentioning

confidence: 88%

A perfect pairing for monoidal adjunctions

Torii¹

2022

Preprint

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Section: Introductionmentioning

confidence: 88%

A perfect pairing for monoidal adjunctions

Torii¹

2022

Preprint

Self Cite

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“…Let us also say immediately that in [23] Lurie already proved that the right adjoint of a strong monoidal functor admits a lax monoidal structure, which suffices for a great many applications. Moreover, Torii has more generally produced a lax monoidal structure (but none of the accompanying coherences) on the right adjoint of an oplax monoidal functor in [29], by means of a span category construction. (We compare his construction to ours in [14].…”

Section: Lax Monoidal Adjunctionsmentioning

confidence: 99%

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

Haugseng,

Hebestreit,

Linskens

et al. 2023

Proceedings of London Math Soc

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We provide a calculus of mates for functors to the ‐category of ‐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 ‐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 ‐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 ‐categories.

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“…In particular, we will extend the explicit description of dual (co)cartesian fibrations to the situation of curved ortho- and Gray fibrations. As an application, we give an explicit description of parametrised adjoints from [HHLN23], extending previous work of Torii [To20].…”

Section: Dualisation and Straightening Of Two-variable Fibrationsmentioning

confidence: 99%

Two-variable fibrations, factorisation systems and -categories of spans

Haugseng,

Hebestreit,

Linskens

et al. 2023

Forum of Mathematics, Sigma

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We prove a universal property for $\infty $ -categories of spans in the generality of Barwick’s adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show that the latter restricts to a self-equivalence on the class of orthogonal adequate triples, which we introduce for this purpose. As applications of the machinery we develop, we give a quick proof of Barwick’s unfurling theorem, show that an orthogonal factorisation system arises from a cartesian fibration if and only if it forms an adequate triple (generalising work of Lanari), extend the description of dual (co)cartesian fibrations by Barwick, Glasman and Nardin to two-variable fibrations, explicitly describe parametrised adjoints (extending work of Torii), identify the orthofibration classifying the mapping category functor of an $(\infty ,2)$ -category (building on work of Abellán García and Stern), formally identify the unstraightenings of the identity functor on the $\infty $ -category of $\infty $ -categories with the (op)lax under-categories of a point, and deduce a certain naturality property of the Yoneda embedding (answering a question of Clausen).

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On Quasi-Categories of Comodules and Landweber Exactness

Cited by 7 publications

References 31 publications

A perfect pairing for monoidal adjunctions

A perfect pairing for monoidal adjunctions

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

Two-variable fibrations, factorisation systems and -categories of spans

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