nLab (infinity,1)-monad

Idea

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

Properties

Barr-Beck monadicity theorem

Proposition

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

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

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

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.

Theorem

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

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

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

3. and $R$ preserves these

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

This appears as (Higher Algebra, theorem 6.2.0.6, theorem 6.2.2.5, Riehl-Verity 13, section 7)

Homotopy coherence

Remark

An (∞,1)-adjunction $(L \dashv R) \colon \mathcal{C} \leftrightarrow \mathcal{D}$ 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 $(\infty,1)$-monads $T \colon \mathcal{C} \to \mathcal{C}$. 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 $(\infty,1)$-monads given via specified (∞,1)-adjunctions as $T \simeq R \circ L$ 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 $\mathcal{D}$.) This should justify the simplicial model category-theoretic discussion in (Hess 10) in (∞,1)-category theory.

Related concepts

• higher monadic descent

• algebraic theory / Lawvere theory / (∞,1)-algebraic theory

• monad / 2-monad/ doctrine / $(\infty,1)$-monad

  • idempotent (∞,1)-monad

  • modal type theory

• operad / (∞,1)-operad

References

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

• Jacob Lurie, section 3 of Noncommutative algebra (math.CT/0702299)

later absorbed as

• Jacob Lurie, section 6.2 of Higher Algebra

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

• Emily Riehl, Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 (arXiv:1310.8279, doi:10.1016/j.aim.2015.09.011)

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

• Kathryn Hess, A general framework for homotopic descent and codescent (arXiv/1001.1556)