Editing Monad (category theory) (section)

===Monadic adjunctions===

Given any adjunction <math>(F : C \to D,G : D \to C,\eta,\varepsilon)</math> with associated monad ''T'', the functor ''G'' can be factored as

:<math>D \overset{\widetilde{G}}\longrightarrow C^T \xrightarrow{\text{forget}} C,</math>

i.e., ''G''(''Y'') can be naturally endowed with a ''T''-algebra structure for any ''Y'' in ''D''. The adjunction is called a '''monadic adjunction''' if the first functor <math>\tilde G</math> yields an [[equivalence of categories]] between ''D'' and the Eilenberg–Moore category <math>C^T</math>.<ref>{{harvtxt|MacLane|1978}} uses a stronger definition, where the two categories are isomorphic rather than equivalent.</ref> By extension, a functor <math>G\colon D\to C</math> is said to be '''monadic''' if it has a left adjoint {{mvar|F}} forming a monadic adjunction.  For example, the free–forgetful adjunction between groups and sets is monadic, since algebras over the associated monad are groups, as was mentioned above. In general, knowing that an adjunction is monadic allows one to reconstruct objects in ''D'' out of objects in ''C'' and the ''T''-action.