Quasi-categories (also called weak Kan complexes and \(\infty\)-categories) were introduced by J. M. Boardman and R. M. Vogt [Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics. 347. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0285.55012)]. Quasi-categories are the fibrant simplicial sets of the Joyal model structure. Quasi-categories can be considered as a generalization of the notion of categories as follows: the nerve of every category is a quasi-category. This generalization is very “solid” in the sense that a lot of theorems of basic category theory and some of the advanced notions and theorems have their analogues in the quasi-categorical context. In this paper, the authors deal with generalizations of theorems due to Schanuel, Street and Beck.
S. Schanuel and R. Street [Cah. Topologie Géom. Différ. Catégoriques 27, No. 1, 81–83 (1986; Zbl 0592.18002)] prove the existence of a \(2\)-category \(\underline{\mathrm{Adj}}\) called the free adjunction such that there exists a bijective correspondence between the \(2\)-functors \(\underline{\mathrm{Adj}} \to \mathcal{K}\) and the adjunctions in \(\mathcal{K}\). The authors prove a homotopy theoretic generalization to the quasi-categorical context which can be stated as follows. They define a cofibrant simplicial category denoted by \(\underline{\mathrm{Adj}}\) and called the free homotopy coherent adjunction such that any low-dimensional adjunction data for an adjunction between quasi-categories extends to a simplicial functor \(\underline{\mathrm{Adj}} \to \underline{\mathrm{qCat}}_\infty\). They then show that suitably defined spaces of all such extensions are contractible. In other terms, any adjunction of quasi-categories extends to a homotopy coherent adjunction, and essentially in a unique way up to homotopy.
The authors extract several simplicial functors from the free homotopy coherent adjunction and show that quasi-categories are closed under weighted limits with these weights. These weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad. They show that each vertex in the quasi-category of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras.
To conclude, the authors prove the quasi-categorical monadicity theorem, describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasi-categories.