Homotopy coherent companionships and conjunctions

Companionships and conjunctions play an important role in the theory of double ∞-categories. Double ∞-categories are two-dimensional ∞-categorical constructions that admit two directions of morphisms. They may be viewed as a generalization of (∞,2)-categories, and, from this perspective, companions and conjoints are the double categorical counterpart of adjoints. We will give examples of companions and conjoints in a range of different contexts. In particular, we will elucidate their fundamental role in a double ∞-categorical approach to formal category theory.

The goal of this talk is to discuss a result that asserts that every companionship/conjunction can uniquely be upgraded to a so-called homotopy coherent one, and if time permits, we will say something about its proof. This result is analogous to the celebrated result by Riehl—Verity that states that adjunctions in (∞,2)-categories upgrade to homotopy coherent adjunctions in a unique way.