Homotopy coherent companionships and conjunctions

TL;DR

This paper develops a homotopy-theoretic framework for understanding companionships and conjunctions in double ∞-categories, proving their unique extensions and characterizing them in functor categories, with applications to (∞,2)-categories.

Contribution

It extends the theory of companionships and conjunctions to functors from free-living structures and characterizes them in functor double Segal spaces, providing new tools for higher category theory.

Findings

Extensions of companionships and conjunctions are homotopically unique.

Characterization of companions and conjoints in functor double Segal spaces.

Application to (∞,2)-category theory.

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.