Homotopy coherent adjunctions and the formal theory of monads

Abstract

Abstract. In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible Kan complexes.We 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. We 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, we 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. Our proofs reveal that a mild variant of Beck's argument is "all in the weights"-much of it independent of the quasi-categorical context.

Data structures for quasistrict higher categories

“…By general results of Riehl and Verity [45], it is expected that such a theorem ought to hold in any reasonable algebraic definition of 4-category. Our proof is the first such that has been given; indeed, we believe it to be the first nontrivial proof in the literature of any sort internal to an algebraic 4-category.…”

Monadicity of the Bousfield–Kuhn functor

“…For background on modules over monads in an ∞-categorical setting, the reader can consult [RV16] or [Lur16]. Assume that G has a left adjoint F , so that there is a corresponding monad M ≃ G • F on C .…”

Free Higher Groups in Homotopy Type Theory

“…Talking about the adjunctions more precisely is difficult since the correct notions would be ∞-categorical. This leads into a territory that is vastly unexplored in homotopy type theory [12], although higher adjunctions can be represented using only a finite amount of data [19]; here, we do not go further into this. Let G be a given ∞-group, represented by (Z, z).…”