The notion of (,1)(\infty,1)-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.


Barr-Beck monadicity theorem


Given an (∞,1)-monad TT on an (∞,1)-category 𝒞\mathcal{C}, there is an (∞,1)-adjunction

(FU):Alg 𝒞(T)UF𝒞, (F \dashv U) \;\colon\; Alg_{\mathcal{C}}(T) \stackrel{\overset{F}{\leftrightarrow}}{\underset{U}{\longrightarrow}} \mathcal{C} \,,

where Alg 𝒞(T)Alg_{\mathcal{C}}(T) is the (Eilenberg-Moore) (∞,1)-category of algebras over the (∞,1)-monad and where UU is the forgetful functor that remembers the underlying object of 𝒞\mathcal{C}.

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 (LR)(L \dashv R) a pair of adjoint (∞,1)-functors such that

  1. RR is a conservative (∞,1)-functor;

  2. the domain (∞,1)-category of RR admits geometric realization ((∞,1)-colimit) of simplicial objects;

  3. and RR preserves these

then for TRLT \coloneqq R \circ L the essentially unique (,1)(\infty,1)-endomorphism monad structure on the composite endofunctor, there is an equivalence of (∞,1)-categories identifying the domain of RR with the (∞,1)-category of algebras over an (∞,1)-monad Alg 𝒞(T)Alg_{\mathcal{C}}(T) over TT and RR itself as the canonical forgetful functor UU from prop. 1.

This appears as (Higher Algebra, theorem, theorem, Riehl-Verity 13, section 7)

Homotopy coherence


An (∞,1)-adjunction (LR):𝒞𝒟(L \dashv R) \colon \mathcal{C} \leftrightarrow \mathcal{D} 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 (,1)(\infty,1)-monads T:𝒞𝒞T \colon \mathcal{C} \to \mathcal{C}. 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 (,1)(\infty,1)-monads given via specified (∞,1)-adjunctions as TRLT \simeq R \circ L are determined by less (further) coherence data (Higher Algebra, remark, prop., Riehl-Verity 13, page 6). (Of course there is, instead, extra data/information carried by the choice of 𝒟\mathcal{D}.) This should justify the simplicial model category-theoretic discussion in (Hess 10) in (∞,1)-category theory.


A general treatment of (,1)(\infty,1)-monads in (∞,1)-category theory is in

later absorbed as

More explict discussion in terms of quasi-categories and simplicial sets is in

Some homotopy theory of (enriched) monads on (simplicial) model categories is discussed (with an eye towards higher monadic descent) in

Revised on May 26, 2014 00:36:10 by Urs Schreiber (