Homotopy coherent adjunctions and the formal theory of monads

By Emily Riehl, Dominic R. Verity. 2013

114 citations recorded by the source corpus.

In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterize 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6bc6ec01f0408b498399633e1ee60b5848c832d4

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Citations in the corpus

The Galois group of a stable homotopy theory (2014, methodology): “This result has recently been reproved from the point of view of weighted (co)limits by Riehl-Verity [RV13].”
Categorical Homotopy Theory (2014, background): “See [RV13b] for a precise statement and proof.”
Fibrations and Yoneda's lemma in an ∞-cosmos (2017, background): “References to [19, 17, 20] will have the form I.”
A type theory for synthetic ∞-categories (2017, background): “As shown in [24], while a complete homotopy-coherent (diagrammatic) adjunction contains infinitely much data, it is uniquely determined by many finite subcollections of data whenever the adjoint relationship exists, such as: a single functor; both functors and the counit; both the unit and counit an”