Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Let be a lax-idempotent 2-monad or pseudo 2-monad on a 2-category , and a pseudo -algebra witnessed by a left adjoint to the unit . We say that is a continuous -algebra if has a further left adjoint, forming an adjoint triple.
Note that every free -algebra is continuous: it is a property of lax-idempotent 2-monads that the multiplication has left adjoint .
For the rest of this page we will say “-algebra” to mean “pseudo -algebra”.
If is the ind-completion monad on Cat, whose algebras are categories with filtered colimits, then a continuous -algebra is precisely a continuous category. This is the origin of the name.
Specializing the previous example, if is the ideal completion monad on Poset, whose algebras are posets with directed joins, then a continuous -algebra is a continuous poset.
Note also that if an adjoint functor theorem applies (such as if and are complete lattices, or locally presentable categories), then is continuous if and only if is a continuous functor. This provides another justification for the name.
For instance, if is the downset monad on Poset, whose algebras are suplattices, then continuous algebras are called constructive completely distributive lattices. Assuming the axiom of choice, these are equivalent to completely distributive lattices. See (WoodFawcett).
If is the free small-cocompletion monad on , whose algebras are cocomplete categories, then continuous algebras are in particular completely distributive categories. The converse only holds if an adjoint functor theorem applies.
If is the presheaf category “monad”, whose “algebras” are total categories, then the “continuous algebras” are totally distributive categories. The scare quotes are because this example is not really a monad for size reasons.
If is the free cocompletion monad under coproducts, then continuous algebras are “locally connected” in a sense: the extra left adjoint decomposes every object as a coproduct of connected ones.
If is a “filter monad” on topological spaces, which can be regarded as a “presheaf category” type construction with topological spaces thought of as generalized multicategories (specifically, as relational beta-modules), then continuous -algebras are called “distributive spaces” in (Hofmann).
Consider the following conditions on a -algebra :
Then (1) (2) (3), and both converses hold if idempotent 2-cells split in the underlying 2-category.
This is B1.1.15 in the Elephant. If the algebra structure has a left adjoint , then is an adjunction in the 2-category of -algebras (since any left adjoint between -algebras is a -morphism). And since the counit of is an isomorphism, the unit of is an isomorphism, by the theorem on fully faithful adjoint triples. This shows (1) (2), and (2) (3) is obvious. For the converse, see the Elephant.
Let be the pseudo 2-comonad on the 2-category of -algebras induced by the free-forgetful adjunction. Then is lax-idempotent, and a -algebra is continuous if and only if it is a -coalgebra.
Lax-idempotence of follows from the fact that the adjunction is a lax-idempotent 2-adjunction. Therefore, the -coalgebras are those -algebras for which the counit has a left adjoint in . However, this counit is just the structure map of , and since is lax-idempotent, any left adjoint between -algebras is automatically a (pseudo) -morphism.
This theorem is proven in (Kock) in the special case when the base 2-category is a (1,2)-category (the result is due to Bart Jacobs).
Last revised on November 22, 2013 at 09:34:29. See the history of this page for a list of all contributions to it.