symmetric monoidal (∞,1)-category of spectra
A functor is monadic iff it has a left adjoint and the adjunction ‘comes from’ the induced monad on – that is, monadic iff is a monadic adjunction.
In this situation in some sense ‘looks like’ the forgetful functor from the Eilenberg-Moore category of the monad on , and has ‘nice properties’ similar to these forgetful functors.
The monadicity theorem characterizes monadic functors and makes these ‘nice properties’ precise.
Given a pair of adjoint functors , , with unit and counit , one constructs a monad setting , .
Consider the Eilenberg-Moore category of -algebras (-modules) in . Clearly is a -action. In fact there is a canonical comparison functor given on objects by . We then say that we have a (resp. strictly) monadic adjunction iff is an equivalence (resp. isomorphism) of categories.
A functor is monadic (resp. strictly monadic) if it has a left adjoint and the comparison functor is an equivalence of categories (resp. an isomorphism of categories). In other words, up to equivalence, monadic functors are precisely the forgetful functors defined on Eilenberg–Moore categories for monads, and strictly monadic functors are the same as these forgetful functors up to isomorphism. A category is monadic over a category if there is a functor which is monadic.
Various versions of Beck’s monadicity theorem (old-fashioned name of some schools: tripleability theorem) give sufficient, and sometimes necessary, conditions for a given functor to be monadic. There are also dual, comonadic versions.
A monadic functor is strictly monadic if and only if it is also an amnestic isofibration: clearly, a strictly monadic functor is an amnestic isofibration; and if a monadic functor is amnestic, then the comparison functor is also amnestic, and if is a monadic isofibration, so is ; therefore in this case must be an isomorphism of categories.
Beware that monadic functors are not closed under composition. For a specific example: the category of reflexive quivers is monadic over via the functor sending a graph to its set of edges, and the category of categories is monadic over reflexive graphs via the forgetful functor , but is not monadic over (via any functor whatsoever, since such categories are regular categories but is not).
This is an instance of a general phenomenon: Let be a reflective subcategory of a presheaf category (e.g. every locally presentable category is of this form). Then the adjunction between and is monadic, and the adjunction between and is also monadic. But the composite adjunction between and is often not monadic. For instance, if it is monadic, then must be a Barr-exact category.
If the comparison functor is only fully faithful, a functor is sometimes said to be of descent type.
Last revised on July 22, 2020 at 05:59:42. See the history of this page for a list of all contributions to it.