Editing Monad (category theory) (section)

====Free-forgetful adjunctions====
For example, let <math>G</math> be the [[forgetful functor]] from [[category of groups|the category '''Grp''']] of [[group (mathematics)|groups]] to the [[category of sets|category '''Set''']] of sets, and let  <math>F</math> be the [[free group]] functor from the category of sets to the category of groups.  Then <math>F</math> is left adjoint of <math>G</math>. In this case, the associated monad <math>T = G \circ F</math> takes a set <math>X</math> and returns the underlying set of the free group <math>\mathrm{Free}(X)</math>.
The unit map of this monad is given by the maps
:<math>X \to T(X) </math>
including any set <math>X</math> into the set <math>\mathrm{Free}(X)</math> in the natural way, as strings of length 1. Further, the multiplication of this monad is the map
:<math>T(T(X)) \to T(X) </math>
made out of a natural [[concatenation]] or 'flattening' of 'strings of strings'.  This amounts to two [[natural transformation]]s.
The preceding example about free groups can be generalized to any type of algebra in the sense of a [[variety of algebras]] in [[universal algebra]].  Thus, every such type of algebra gives rise to a monad on the category of sets.  Importantly, the algebra type can be recovered from the monad (as the category of Eilenberg–Moore algebras), so monads can also be seen as generalizing varieties of universal algebras.

Another monad arising from an adjunction is when <math>T</math> is the endofunctor on the category of vector spaces which maps a vector space <math>V</math> to its [[tensor algebra]] <math>T(V)</math>, and which maps linear maps to their tensor product. We then have a natural transformation corresponding to the embedding of <math>V</math> into its [[tensor algebra]], and a natural transformation corresponding to the map from <math>T(T(V))</math> to <math>T(V)</math> obtained by simply expanding all tensor products.