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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 “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.
Monadic functors are sometimes called functors of effective descent type. See the page on monadic descent for more on this aspect.
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.
(monadic functor)
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 strict 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 called monadic over a category if there is any functor which is monadic (Def. ).
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 the class of monadic functors is not closed under composition.
For a specific counter-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.
Various versions of Beck’s monadicity theorem (also: “tripleability theorem” in older literature) give sufficient, and sometimes necessary, conditions for a given functor to be monadic. There are also dual, comonadic versions.
Francis Borceux, Def. 4.4.1 in: Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]
Emily Riehl, §5.3 in: Category Theory in Context, Dover Publications (2017) [pdf]
Last revised on November 6, 2022 at 10:50:32. See the history of this page for a list of all contributions to it.