Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
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Algebras and modules
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Geometry on formal duals of algebras
The Eilenberg–Moore (EM) category of a monad is the category of its modules (aka algebras). Dually, the EM category of a comonad is its category of comodules. The subcategory of its free modules is one of the descriptions of the Kleisli category of the monad. The EM and Kleisli categories have universal properties which make sense in a general 2-category.
Let be a monad in Cat, where is an endofunctor with multiplication and unit . Recall that a (left) -module (or -algebra) in is a pair of an object in and a morphism which is a -action, namely and , and that a morphism of -modules is a morphism in that commutes with the action: . The composition of morphisms of -modules is the composition of underlying morphisms in .
-modules and their morphisms thus form a category which is called the Eilenberg–Moore category of the monad . This may also be written , , etc. It comes equipped with a forgetful functor which is the universal -module, and has a left adjoint such that the monad arising from the adjunction is equal to .
In general, if is a monad in a 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.
If is the Kleisli category of and the canonical functor, then the EM category can be constructed as the pullback
Thus a -algebra may be regarded as a presheaf on the Kleisli category of whose restriction to is representable. This observation seems to be due to Linton. Street–Walters show that it holds in any 2-category equipped with a Yoneda structure?.
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.
By lax 2-limits
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, JPAA 2, 1972
- Fred Linton, An outline of functorial semantics, in LNM 80, 1969
- Ross Street, Bob Walters, Yoneda structures, J. Algebra 50, 1978
Local presentability of EM-categories is discussed on p. 123, 124 of
The following paper of Melliès compares the Linton representability condition above 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