Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The category of algebras over a monad (also: “modules over a monad”) is traditionally called its Eilenberg–Moore category (EM). Dually, the EM category of a comonad is its category of coalgebras (co-modules).
The subcategory of (co-)free (co-)algebras is traditionally called the Kleisli category of the (co-)monad.
The EM and Kleisli categories have universal properties which make sense for (co-)monads in any 2-category (not necessarily Cat).
Let be a monad in Cat, where is an endofunctor with multiplication and unit .
A (left) -module (or -algebra) in is a pair of an object in and a morphism which is a -action, in that
A homomorphism of -modules is a morphism in that commutes with the action, in that
The composition of morphisms of -modules is the composition of underlying morphisms in . The resultiing category of -modules/algebras is called the Eilenberg–Moore category of the monad , also be written , or , etc.
By construction, there is a forgetful functor
(which may be thought of as the universal -module) with a left adjoint free functor such that the monad arising from the adjunction is isomorphic to .
More generally, for is a monad in any 2-category , then the Eilenberg–Moore object of is, if it exists, the universal (left) -module. That is, there is a morphism and a 2-cell that mediate a natural isomorphism between morphisms and -modules . Not every 2-category admits Eilenberg–Moore objects.
Apart from being the universal left -module, the EM category of a monad in has some other interesting properties.
There is a full subcategory of the slice category on the functors that have left adjoints. For any monad on there is a full subcategory of this consisting of the adjoint pairs that compose to give . The functor is the terminal object of this category.
As a colimit completion of the Kleisli category
Every -algebra is the coequalizer of the first stage of its bar resolution:
This is a reflexive coequalizer of -algebras. Moreover, the underlying fork in is a split coequalizer, hence in particular an absolute coequalizer (sometimes called the Beck coequalizer, due to its role in the Beck monadicity theorem). A splitting is given by
(e.g. MacLane, bottom of p. 148 and exercise 4 on p. 151) See also at split coequallizer – Beck coequalizer for algebras over a monad.
In particular this says that every -algebra is presented by free -algebras. The nature of -algebras as a kind of completion of free -algebras under colimits is made more explicit as follows.
Write for the Kleisli category of , the category of free -algebras. Write the free functor. Observe that via the inclusion every -algebra represents a presheaf on . Recall that the category of presheaves is the free cocompletion of .
The -algebras in are equivalently those presheaves on the category of free -algebras whose restriction along the free functor is representable in . In other words, the Eilenberg-Moore category is the (1-category theoretic) pullback
of the category of presheaves on the Kleisli category along the Yoneda embedding of .
This statement appears as (Street 72, theorem 14). It seems to go back to (Linton 69), see (Melliès 10, p. 4). (Street-Walters 78) show that it holds in any 2-category equipped with a Yoneda structure?
By lax 2-limits
Just as the Kleisli object of a monad in a 2-category can be defined as the lax colimit of the lax functor corresponding to , the EM object of is its lax limit.
S. Lack has shown how Eilenberg-Moore objects can be obtained as combinations of certain simpler lax limits, when the 2-category in question is the 2-category of 2-algebras over a 2-monad and lax, colax or pseudo morphisms of such:
- Steve Lack, Limits for lax morphisms , Applied Categorical Structures 13:3 (2005) , pp. 189–203(15)
This encompasses for example the theory of (op)monoidal monads and corresponding monoidal Eilenberg–Moore categories.
If is a monad in a small category , and is another category, then consider the functor category . There is a tautological monad on defined by , , , , , . Then there is a canonical isomorphism of EM categories
Namely, write the object part of a functor as , where and is the -action of and the morphism part simply as . Then, is a natural transformation because for any morphism , is by the definition of , a morphism of -algebras. is, by the same argument, an action . Conversely, for any -module for any , will evaluate to a -action on , hence is an object part of a functor in with morphism part again . The correspondence for the natural transformations, is similar.
Dually, for a comonad in , there is a canonical comonad on and an isomorphism of categories
Limits and colimits in EM categories
The Eilenberg-Moore category of a monad on a category has all limits which exist in , and they are created by the forgetful functor.
In contrast, the subject of colimits in categories of algebras is less easy, but a good deal can be said.
Moreover, let be a topos. Then
if a monad has a right adjoint then is itself a topos;
if a comonad is left exact, then is itself a topos.
See at topos of algebras over a monad for details.
Given a reflective subcategory then the Eilenberg-Moore category of the induced idempotent monad on recovers the subcategory .
For instance (Borceux, vol 2, cor. 4.2.4).
General discussion is in
Ross Street, The formal theory of monads, Journal of Pure and Applied Algebra 2, 1972
Fred Linton, An outline of functorial semantics, in LNM 80, 1969
Fred Linton, Relative functorial semantics: adjointness results, Lecture Notes in Mathematics, vol. 99, 1969
Ross Street, Bob Walters, Yoneda structures, J. Algebra 50, 1978
Saunders MacLane, Categories for the Working Mathematician
Local presentability of EM-categories is discussed on p. 123, 124 of
The following paper of Melliès compares the representability condition of (Linton 69) with the Segal condition that distinguishes those simplicial sets that are the nerves of categories.
The example of idempotent monads is discussed also in
Discussion for (infinity,1)-monads realized in the context of quasi-categories is around def. 6.1.7 of