symmetric monoidal (∞,1)-category of spectra
The notion of -monad is the vertical categorification of that of monad from the context of categories to that of (∞,1)-categories.
They relate to (∞,1)-adjunctions as monads relate to adjunctions.
Given an (∞,1)-monad on an (∞,1)-category , there is an (∞,1)-adjunction
where is the (Eilenberg-Moore) (∞,1)-category of algebras over the (∞,1)-monad and where is the forgetful functor that remembers the underlying object of .
This appears in (Riehl-Verity 13, def. 6.1.14).
The following is the refinement to (∞,1)-category theory of the classical Barr-Beck monadicity theorem which states sufficient conditions for recognizing an (∞,1)-adjunction as being canonically equivalent to the one in prop. , hence to be a monadic adjunction.
Let a pair of adjoint (∞,1)-functors such that
the domain (∞,1)-category of admits geometric realization ((∞,1)-colimit) of simplicial objects;
and preserves these
then for the essentially unique -monad structure on the composite endofunctor, there is an equivalence of (∞,1)-categories identifying the domain of with the (∞,1)-category of algebras over an (∞,1)-monad over and itself as the canonical forgetful functor from prop. .
This appears as (Higher Algebra, theorem 6.2.0.6, theorem 6.2.2.5, Riehl-Verity 13, section 7)
An (∞,1)-adjunction is uniquely determined already by its image in the homotopy 2-category (Riehl-Verity 13, theorem 5.4.14). This is not in general true for -monads . As these are monoids in an (∞,1)-category of endomorphisms, they in general have relevant coherence data all the way up in degree. However, by the previous statement and the monadicity theorem , for -monads given via specified (∞,1)-adjunctions as are determined by less (further) coherence data (Higher Algebra, remark 6.2.0.7, prop. 6.2.2.3, Riehl-Verity 13, page 6). (Of course there is, instead, extra data/information carried by the choice of .) This should justify the simplicial model category-theoretic discussion in (Hess 10) in (∞,1)-category theory.
A general treatment of -monads in (∞,1)-category theory is in
later absorbed as
More explict discussion in terms of quasi-categories and simplicial sets:
Some homotopy theory of (enriched) monads on (simplicial) model categories is discussed (with an eye towards higher monadic descent) in
Last revised on September 22, 2020 at 13:58:49. See the history of this page for a list of all contributions to it.