symmetric monoidal (∞,1)-category of spectra
This appears in (Riehl-Verity 13, def. 6.1.15).
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. 1, hence to be a monadic adjunction.
Let a pair of adjoint (∞,1)-functors such that
and preserves these
then for the essentially unique -endomorphism 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. 1.
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 1, for -monads given via specified (∞,1)-adjunctions as are determined by less (further) coherence data (Higher Algebra, remark 18.104.22.168, prop. 22.214.171.124, 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