Seminar

There are three useful ways to think about a monad on an ∞-category C: as an associative algebra in endofunctors of C, as a monadic right adjoint functor to C, and as a functor from the universal monad 2-category to the (∞,2)-category of ∞-categories. These notions are known to be equivalent, thanks to work of Lurie and Riehl-Verity - but only for ∞-groupoids of monads. I will explain how to upgrade this comparison to take morphisms of monads into account. One comparison involves a general equivalence between colax morphisms of adjunctions in an (∞,2)-category and commutative squares between right adjoints, which can be proved using the expected properties of the Gray tensor product, together with results of Riehl-Verity and Zaganidis. The other (which I will probably say less about) requires comparing the natural transformations between (∞,2)-categories viewed as cocartesian fibrations over the simplex category to certain colax transformations (generalizing the "icons" of Lack).