Kleisli Constructions on Simplicial sets and Quasi-categories

Adrian Miranda – 8 September 2021

A monad can be defined on a simplicial set X in a way which specialises to the usual definition when X is the nerve of a category. Then the weights for Kleisli and Eilenberg-Moore constructions can also be transported along N: Cat --> sSet to the simplicially enriched setting. This perspective is exploited by Reihl and Verity to define homotopy coherent monads on quasicategories (and indeed in infinity cosmoses). In the first part of the talk, we will simplify the rather abstract notion of Kleisli objects to a more intuitive 'coequaliser of monoid actions'. We will give a concrete and elementary, if somewhat strange, description of the simplicially enriched Kleisli object of a monad on a simplicial set. We will see that, unlike in the Cat-enriched setting, even if we start off with an identity monad on the terminal simplicial set, we get something seemingly not as trivial out of this construction.

In the second part of the talk, we will restrict our attention to Kleisli constructions for monads on quasi-categories. Guided by our intuition that the Kleisli object should be 'free algebras and arbitrary algebra morphisms', we will consider the essentially surjective on objects-fully faithful factorisation of the free functor of the Eilenberg-Moore adjunction. I will outline a proof for why this is a Kleilsi object for the monad, not in the sSet enriched sense but in the 'up to weak equivalence' sense. The key ingredient in our analysis will be Riehl & Verity's generalisation of Beck's monadicity theorem to the quasi-categorical setting. Our main result will be a direct generalisation to the quasi-categories setting of the standard fact that a left adjoint functor is Kleisli in the up to equivalence sense if and only if it is essentially surjective on objects. I will conclude by sharing some musings on how this might be generalised from the qCat setting to the infinity cosmos setting, possibly under the assumption of some extra structure.