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

Rune Haugseng, Fabian Hebestreit, Sil Linskens, Joost Nuiten

Abstract

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.

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

Abstract

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.

Paper Structure

This paper contains 19 sections, 70 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Two-variable fibrations
  3. Background
  4. Straightening in one variable
  5. Curved orthofibrations
  6. Gray fibrations
  7. Dualisation of curved orthofibrations and Gray fibrations
  8. Parametrised and monoidal adjunctions
  9. Parametrised adjunctions
  10. Identifying mates
  11. Parametrised correspondences
  12. Lax monoidal adjunctions
  13. Parametrised units and counits
  14. Parametrised (co)units
  15. Passing to adjoint morphisms
  16. ...and 4 more sections

Key Result

Proposition A

Given two symmetric monoidal $\infty$-categories $\StrLen{C}[\mystrlen] \mathrm{C}$ and $\StrLen{D}[\mystrlen] \mathrm{D}$, the extraction of adjoints gives inverse equivalences between the $\infty$-category of lax symmetric monoidal right adjoints $\StrLen{C}[\mystrlen] \mathrm{C} ^{\otimes} \right

Theorems & Definitions (160)

  • Proposition A
  • Theorem B
  • Corollary C
  • Theorem D
  • Theorem E
  • Corollary F
  • Remark
  • Definition 2.1.1
  • Definition 2.1.3
  • Remark 2.1.5
  • ...and 150 more