Editing Monad (category theory) (section)

===Beck's monadicity theorem===

''[[Beck's monadicity theorem]]'' gives a necessary and sufficient condition for an adjunction to be monadic. A simplified version of this theorem states that ''G'' is monadic if it is [[conservative functor|conservative]] (or ''G'' reflects isomorphisms, i.e., a morphism in ''D'' is an isomorphism if and only if its image under ''G'' is an isomorphism in ''C'') and ''C'' has and ''G'' preserves [[coequalizer]]s.

For example, the forgetful functor from the category of [[compact topological space|compact]] [[Hausdorff space]]s to sets is monadic. However the forgetful functor from all topological spaces to sets is not conservative since there are continuous bijective maps (between non-compact or non-Hausdorff spaces) that fail to be [[homeomorphisms]]. Thus, this forgetful functor is not monadic.<ref>{{harvtxt|MacLane|loc=§§VI.3, VI.9|1978}}</ref>
The dual version of Beck's theorem, characterizing comonadic adjunctions, is relevant in different fields such as [[topos theory]] and topics in [[algebraic geometry]] related to [[descent (category theory)|descent]]. A first example of a comonadic adjunction is the adjunction

:<math>- \otimes_A B : \mathbf{Mod}_A \rightleftarrows \mathbf{Mod}_B : \operatorname{forget}</math>

for a ring homomorphism <math>A \to B</math> between commutative rings. This adjunction is comonadic, by Beck's theorem, if and only if ''B'' is [[Faithfully flat module|faithfully flat]] as an ''A''-module. It thus allows to descend ''B''-modules, equipped with a descent datum (i.e., an action of the comonad given by the adjunction) to ''A''-modules. The resulting theory of [[faithfully flat descent]] is widely applied in algebraic geometry.