nLab module over a monad



2-Category theory

Higher algebra



Just as the notion of a monad in a bicategory KK generalizes that of a monoid in a monoidal category, modules over monoids generalize easily to modules over monads.

Beware that modules over monads in Cat are often called *algebras* for the monad (see there for more), since they literally are algebras in the sense of universal algebra, see below. By extension, one might speak of modules over monads in any 2-category as “algebras for the monad”.

The formally dual concept is that of coalgebra over a comonad.



Let KK be a bicategory and t:aat \colon a \to a a monad in KK with structure 2-cells μ:ttt\mu \colon t t \Rightarrow t and η:1 at\eta \colon 1_a \Rightarrow t. Then a left tt-module (or tt-algebra) is given by a 1-cell x:bax \colon b \to a and a 2-cell λ:txx\lambda \colon t x \Rightarrow x, where

ttx μx tx tλ λ tx λ xx ηx tx 1 λ x \array{ t t x & \overset{\mu x}{\to} & t x \\ t\lambda\downarrow & & \downarrow \lambda \\ t x & \underset{\lambda}{\to} & x } \qquad \qquad \array{ x & \overset{\eta x}{\to} & t x \\ & 1\searrow & \downarrow \lambda \\ & & x }

commute. Similarly, a right tt-module (or tt-opalgebra) is given by a 1-cell y:acy \colon a \to c and a 2-cell ρ:yty\rho \colon y t \Rightarrow y, with commuting diagrams as above with yy on the left instead of xx on the right.

Clearly, a right tt-module in KK is the same thing as a left tt-module in K opK^{\mathrm{op}}. A left tt-comodule or coalgebra is then a left tt-module in K coK^{\mathrm{co}}, and a right tt-comodule is a left tt-module in K coopK^{\mathrm{coop}}.

A tt-module of any of these sorts is a fortiori an algebra over the underlying endomorphism tt.


Given monads ss on bb and tt on aa, an s,ts,t-bimodule is given by a 1-cell x:bax\colon b \to a, together with the structures of a right ss-module ρ:xsx\rho \colon x s \Rightarrow x and a left tt-module λ:txx\lambda \colon t x \Rightarrow x that are compatible in the sense that the diagram

txs tρ tx λs λ xs ρ x \array{ t x s & \overset{t\rho}{\to} & t x \\ \lambda s \downarrow & & \downarrow \lambda \\ x s & \underset{\rho}{\to} & x }

commutes. Such a bimodule may be written as x:stx \colon s ⇸ t.

A morphism of left tt-modules (x,λ)(x,λ)(x,\lambda) \to (x',\lambda') is given by a 2-cell α:xx\alpha \colon x \Rightarrow x' such that λtα=αλ\lambda' \circ t\alpha = \alpha \circ \lambda. Similarly, a morphism of right tt-modules (y,ρ)(y,ρ)(y,\rho) \to (y',\rho') is β:yy\beta \colon y \Rightarrow y' such that ραs=αρ\rho' \circ \alpha s = \alpha \circ \rho. A morphism of bimodules (x,λ,ρ)(x,λ,ρ)(x,\lambda,\rho) \to (x',\lambda',\rho') is given by α:xx\alpha \colon x \Rightarrow x' that is a morphism of both left and right modules.

More abstractly, the monads ss and tt in KK give rise to ordinary monads s *s^* and t *t_* on the hom-category K(b,a)K(b,a), by pre- and post-composition. The associativity isomorphism of KK then gives rise to an invertible distributive law between these, so that the composite s *t *t *s *:xtxss^* t_* \cong t_* s^* \colon x \mapsto t x s is again a monad. Then the category Mod K(s,t)Mod_K(s,t) of bimodules from ss to tt is the ordinary Eilenberg–Moore category K(b,a) s *t *K(b,a)^{s^* t_*}.

Algebras for monads in Cat

If K=K = Cat and (T,η,μ)(T,\eta,\mu) is a monad on a category CC, then a left TT-module A:1CA \colon 1 \to C, where 11 is the terminal category, is usually called an algebra over T T or TT-algebra (see there): it is given by an object ACA \in C together with a morphism α:TAA\alpha \colon T A \to A, such that

T(T(A)) μ A T(A) T(α) α T(A) α A \array { T(T(A)) & \stackrel{\mu_A}\rightarrow & T(A) \\ T(\alpha) \downarrow & & \downarrow \alpha \\ T(A) & \stackrel{\alpha}\rightarrow & A }


A η A T(A) id A α A \array { A & \stackrel{\eta_A}\rightarrow & T(A) \\ & id_A \searrow & \downarrow \alpha \\ & & A }


In particular, every algebra over a monad (T,η,μ)(T,\eta,\mu) in CatCat has the structure of an algebra over the underlying endofunctor TT.

TT-algebras can also be defined as left modules over TT qua monoid in End(C)End(C). There the object AA is represented by the constant endofunctor at AA.

The Eilenberg-Moore category of TT is the category of these algebras. It has a universal property that allows the notion of Eilenberg-Moore object to be defined in any bicategory.

In a virtual double category

It’sthe notion of (bi)module makes sense in virtual double categories, generalizing the previous definition.

A monad in a virtual double category is a loose endomorphism t:aat : a \nrightarrow a together with a binary map (i.e.~a square in the virtual double category) μ:t,tt\mu:t,t \to t giving the multiplication and a nullary map η:()t\eta:() \to t giving the unit:

This data satisfies strict unitality and associativity equations.

A left module mm over a monad t:aat:a \nrightarrow a is another loose cell m:abm:a \nrightarrow b together with a binary map :t,mm\rhd : t,m \to m:

This again satisfies standard equations stating compatibility with the monad structure on tt.

A right module is equipped with a binary map :m,tm\lhd : m, t \to m instead, and a bimodule has a ternary map :t,m,tm\rhd - \lhd : t,m,t \to m.



see at colimits in categories of algebras

Tensor product

Given bimodules x:rsx' \colon r ⇸ s and x:stx \colon s ⇸ t, where r,s,tr,s,t are monads on c,b,ac,b,a respectively, we may be able to form the tensor product x sx:rtx \otimes_s x' \colon r ⇸ t just as in the case of bimodules over rings. If the hom-categories of the bicategory KK have reflexive coequalizers that are preserved by composition on both sides, then the tensor product is given by the reflexive coequalizer in K(c,a)K(c,a)

xsx xx x sx \array{ x s x' & \overset{\to}{\to} & x x' & \to x \otimes_s x' }

where the parallel arrows are the two induced actions ρx\rho x' and xλx \lambda on ss. Indeed, under the hypothesis on KK the forgetful functor Mod K(r,t)=K(c,a) r *t *K(c,a)Mod_K(r,t) = K(c,a)^{r^* t_*} \to K(c,a) reflects reflexive coequalizers, because the monad r *t *r^* t_* preserves them, and so x sxx \otimes_s x' is an r,tr,t-bimodule.

If KK satisfies the above conditions then there is a bicategory Mod(K)Mod(K) consisting of monads, bimodules and bimodule morphisms in KK. The identity module on a monad tt is tt itself, and the unit and associativity conditions follow from the universal property of the above coequalizer. There is a lax forgetful functor Mod(K)KMod(K) \to K, with comparison morphisms 1 at1_a \to t the unit of tt, and xxx sxx x' \to x \otimes_s x' the coequalizer map.


If K=Span(Set)K = Span(Set), the bicategory of spans of sets, then a monad in KK is precisely a small category. Then Mod(K)=ProfMod(K) = Prof, the category of small categories, profunctors and natural transformations.

More generally, Mod(Span(C))Mod(Span(C)), for CC any category with coequalizers and pullbacks that preserve them, consists of internal categories in CC, together with internal profunctors between them and transformations between those.


  • John Isbell, Generic algebras Transactions of the AMS, vol 275, number 2 (pdf)

  • H. Lindner, Commutative monads in Deuxiéme colloque sur l’algébre des catégories. Amiens-1975. Résumés des conférences, pages 283-288. Cahiers de topologie et géométrie différentielle catégoriques, tome 16, nr. 3, 1975.

  • R. Guitart, Tenseurs et machines, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21(1):5-62, 1980.

  • A. Kock. Closed categories generated by commutative monads, Journal of the Australian Mathematical Society, 12(04):405-424, 1971.

  • G. J. Seal. Tensors, monads and actions, Theory Appl. Categ., 28:No. 15, 403-433, 2013.

Discussion of model category structures on categories of coalgebras over comonads is in

Last revised on November 15, 2023 at 11:28:04. See the history of this page for a list of all contributions to it.