Editing Monad (category theory) (section)

==== Commutative algebras in E-infinity ring spectra ====
There is an analogous construction for [[S-module|commutative <math>\mathbb{S}</math>-algebras]]<ref>{{Cite journal|date=1999-12-15|title=André–Quillen cohomology of commutative S-algebras|journal=[[Journal of Pure and Applied Algebra]]|language=en|volume=144| issue=2|pages=111–143| doi=10.1016/S0022-4049(98)00051-6|issn=0022-4049|doi-access=free|last1=Basterra |first1=M. }}</ref><sup>pg 113</sup> which gives commutative <math>A</math>-algebras for a commutative <math>\mathbb{S}</math>-algebra <math>A</math>. If <math>\mathcal{M}_A</math> is the category of <math>A</math>-modules, then the functor <math>\mathbb{P}: \mathcal{M}_A \to \mathcal{M}_A</math> is the monad given by<math display="block">\mathbb{P}(M) = \bigvee_{j \geq 0} M^j/\Sigma_j</math>where<math display="block">M^j = M\wedge_A \cdots \wedge_A M</math> <math>j</math>-times. Then there is an associated category <math>\mathcal{C}_A</math> of commutative <math>A</math>-algebras from the category of algebras over this monad.