Monads and monadicity

An adjunction \(F\dashv G\) determines the endofunctor \(GF\) together with a unit and multiplication. In classical category theory these maps satisfy the monad identities strictly. For \(\infty \)-categories, they must be accompanied by coherent higher homotopies, so it is preferable to package a monad as an algebra object rather than to list this structure by hand. The Barr–Beck theorem then recognizes when the target of the adjunction is recovered as the \(\infty \)-category of modules over this monad.

We use Lurie’s construction of the coherent monad associated to an adjunction and its Eilenberg–Moore category, [Lurie (2017), Section 4.7]. The recognition criterion follows from his comparison theorem, recorded as Theorem 21.7.3 in Part III.

We first construct the monoidal structure under which monads are algebras, as an instance of the endomorphism-algebra construction of Section 19.4. Regard \(\Cat _{\infty }\) as left-tensored over itself via the cartesian product, and fix the object \(C\). By the currying axiom for functor categories (Axiom C.4), the functor category \(\Fun (C,C)\), equipped with the evaluation functor \(\ev \colon \Fun (C,C) \times C \to C\), is an endomorphism object of \(C\) in \((\Cat _{\infty },\times )\): precomposition with \(\ev \) induces an isomorphism \[ \Hom _{\Cat _{\infty }}(X,\Fun (C,C)) \iso \Hom _{\Cat _{\infty }}(X \times C,C) \] for every \(X \in \Cat _{\infty }\). Applying Corollary 19.4.7 therefore equips \(\Fun (C,C)\) with a preferred associative algebra structure in \((\Cat _{\infty },\times )\). Under the simplicial comparisons of Theorem 19.3.2, Theorem 19.3.5, this is a monoidal structure whose tensor product \(S \otimes T \simeq S \circ T\) is given by composition, and the resulting left module structure exhibits \(C\) as left-tensored over \(\Fun (C,C)\) through evaluation. When \(C\) is not small we run this construction in a larger universe, exactly as in Proposition 19.5.5.

Informally, the algebra structure consists of a unit \(\id _C \to T\) and a multiplication \(T \circ T \to T\), together with all coherent associativity and unitality data. A left \(T\)-module is an object \(c \in C\) with a coherent action \(T(c) \to c\).

We record the part of Lurie’s package needed below. Given an adjunction \[ F\colon C \rightleftarrows D \noloc G, \] the composite \(GF\) admits a preferred monad structure \(T\), characterized as the endomorphism algebra of \(G\). The resulting \(T\)-action on \(G\) determines a comparison functor \[ K_G\colon D \longrightarrow \LMod _T(C) \] with \(U_TK_G \simeq G\); here \(\Fun (D,C)\) is left-tensored over \(\Fun (C,C)\) by postcomposition, which is the action used to form the endomorphism algebra of \(G\). See [Lurie (2017), Definition 4.7.3.2, Proposition 4.7.3.3 and the discussion preceding Definition 4.7.3.4]. The forgetful functor \(U_T\) is conservative and admits a left adjoint \(F_T\), whose underlying endofunctor is \(T\). The mate \[ F_T \longrightarrow K_GF \] is an isomorphism. Finally, every \(U_T\)-split simplicial object of \(\LMod _T(C)\) admits a geometric realization which is preserved by \(U_T\); see [Lurie (2017), Corollary 4.2.3.2, Corollary 4.2.4.8, Lemma 4.7.3.12].

The basic example is the forgetful functor from modules. The following result makes precise the idea that an \(A\)-module is an algebra for the operation of tensoring with \(A\).

Applied to the monad \(A\otimes -\) of Proposition 19.6.5, part (2) shows that the forgetful functor \(\LMod _A(D)\to D\) creates any colimits which exist in \(D\) and are preserved by \(A\otimes -\). For a fixed algebra this sharpens Corollary 19.1.17(2), whose more uniform hypothesis ensures the same conclusion simultaneously for every algebra in \(C\).