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Homotopy coherent companionships and conjunctions

Jaco Ruit

TL;DR

The paper develops a homotopy-coherent theory of companionships and conjoints in double ∞-categories via double Segal spaces. It proves that every companionship extends to a homotopy coherent one with contractible extension spaces and provides a two-sided characterization of conjoints, including a strong relation to adjunction data. It then applies these results to functor double Segal spaces to characterize companions/conjoints through companionable and conjointable 2-cells, and constructs double ∞-categories of lax functors, along with a Gray tensor product framework. The approach yields a conceptual, universal treatment of adjunction-like phenomena in higher category theory and supplies tools for (∞,2)-categorical applications, notably in lax natural transformations and functorial Gray tensorial structures. Overall, the work connects extension-universality, homotopy coherence, and higher functor theory to produce a robust foundation for (∞,2)-categorical adjunctions and their representations.

Abstract

We demonstrate that companionships and conjunctions in double $\infty$-categories -- and more generally, in double Segal spaces -- extend to functors out of the free-living companionship and conjunction respectively. Specifically, we prove that these extensions are (homotopically) unique: the corresponding spaces of extensions are contractible under suitable completeness assumptions. The developed theory is then put to use to give a characterization of companions and conjoints in functor double Segal spaces in terms of so-called companionable and conjointable 2-cells. We end with an application of our results to $(\infty,2)$-category theory.

Homotopy coherent companionships and conjunctions

TL;DR

The paper develops a homotopy-coherent theory of companionships and conjoints in double ∞-categories via double Segal spaces. It proves that every companionship extends to a homotopy coherent one with contractible extension spaces and provides a two-sided characterization of conjoints, including a strong relation to adjunction data. It then applies these results to functor double Segal spaces to characterize companions/conjoints through companionable and conjointable 2-cells, and constructs double ∞-categories of lax functors, along with a Gray tensor product framework. The approach yields a conceptual, universal treatment of adjunction-like phenomena in higher category theory and supplies tools for (∞,2)-categorical applications, notably in lax natural transformations and functorial Gray tensorial structures. Overall, the work connects extension-universality, homotopy coherence, and higher functor theory to produce a robust foundation for (∞,2)-categorical adjunctions and their representations.

Abstract

We demonstrate that companionships and conjunctions in double -categories -- and more generally, in double Segal spaces -- extend to functors out of the free-living companionship and conjunction respectively. Specifically, we prove that these extensions are (homotopically) unique: the corresponding spaces of extensions are contractible under suitable completeness assumptions. The developed theory is then put to use to give a characterization of companions and conjoints in functor double Segal spaces in terms of so-called companionable and conjointable 2-cells. We end with an application of our results to -category theory.
Paper Structure (23 sections, 45 theorems, 189 equations, 2 tables)

This paper contains 23 sections, 45 theorems, 189 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Double Segal spaces and the cartesian closed structure
  3. Companionships and conjunctions
  4. Companions of vertical natural transformations
  5. An application to $(\infty,2)$-category theory
  6. Preliminaries on exponentiability
  7. Preliminaries on double Segal spaces
  8. Double Segal spaces
  9. Two-fold Segal spaces
  10. Horizontal and vertical fragments
  11. Completeness conditions
  12. Exponentiability of double Segal spaces
  13. The squares construction
  14. Companionships and conjunctions, and homotopy coherence
  15. Homotopy coherence
  16. ...and 8 more sections

Key Result

Theorem A

Any companionship in a double Segal space $\mathscr{P}$ extends to a homotopy coherent companionship $\mathfrak{comp} \rightarrow \mathscr{P}$. Moreover, suppose that $\eta$ is a companionship unit. Then the space of functors $\mathfrak{comp} \rightarrow \mathscr{P}$ extending $\eta$ is contractible

Theorems & Definitions (145)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Example 1
  • Example 2
  • ...and 135 more