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. 6.1.14).

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 (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)-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. .

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 , 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:

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 09:58:49. See the history of this page for a list of all contributions to it.