symmetric monoidal (∞,1)-category of spectra
Modules over monads, especially in Cat, are also often called algebras for the monad; see below.
Let be a bicategory and a monad in with structure 2-cells and . Then a left -module is given by a 1-cell and a 2-cell , where
commute. Similarly, a right -module is given by a 1-cell and a 2-cell , with commuting diagrams as above with on the left instead of on the right.
Clearly, a right -module in is the same thing as a left -module in . A left -comodule or coalgebra is then a left -module in , and a right -comodule is a left -module in .
A -module of any of these sorts is a fortiori an algebra over the underlying endomorphism .
Given monads on and on , an -bimodule is given by a 1-cell , together with the structures of a right -module and a left -module that are compatible in the sense that the diagram
commutes. Such a bimodule may be written as .
A morphism of left -modules is given by a 2-cell such that . Similarly, a morphism of right -modules is such that . A morphism of bimodules is given by that is a morphism of both left and right modules.
More abstractly, the monads and in give rise to ordinary monads and on the hom-category , by pre- and post-composition. The associativity isomorphism of then gives rise to an invertible distributive law between these, so that the composite is again a monad. Then the category of bimodules from to is the ordinary Eilenberg--Moore category .
If and is a monad on a category , then a left -module , where is the terminal category, is usually called a -algebra: it is given by an object together with a morphism , such that
In particular, every algebra over a monad in has the structure of an algebra over the underlying endofunctor .
-algebras can also be defined as left modules over qua monoid in . There the object is represented by the constant endofunctor at .
Given bimodules and , where are monads on respectively, we may be able to form the tensor product just as in the case of bimodules over rings. If the hom-categories of the bicategory have reflexive coequalizers that are preserved by composition on both sides, then the tensor product is given by the reflexive coequalizer in
where the parallel arrows are the two induced actions and on . Indeed, under the hypothesis on the forgetful functor reflects reflexive coequalizers, because the monad preserves them, and so is an -bimodule.
If satisfies the above conditions then there is a bicategory consisting of monads, bimodules and bimodule morphisms in . The identity module on a monad is itself, and the unit and associativity conditions follow from the universal property of the above coequalizer. There is a lax forgetful functor , with comparison morphisms the unit of , and the coequalizer map.
More generally, , for any category with coequalizers and pullbacks that preserve them, consists of internal categories in , together with internal profunctors between them and transformations between those.
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.