Editing Monad (category theory) (section)

== Algebras for a monad ==
{{see also|F-algebra|pseudoalgebra}}

Given a monad <math>(T,\eta,\mu)</math> on a category <math>C</math>, it is natural to consider ''<math>T</math>-algebras'', i.e., objects of <math>C</math> acted upon by <math>T</math> in a way which is compatible with the unit and multiplication of the monad. More formally, a <math>T</math>-algebra  <math>(x,h)</math> is an object <math>x</math> of <math>C</math> together with an arrow <math>h\colon Tx\to x</math> of <math>C</math> called the ''structure map'' of the algebra such that the diagrams
{| align="center"
|-
|[[Image:Monad_multi_algebra.svg|class=skin-invert]]
| width="20" |
|and
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|[[Image:Monad_unit_algebra.svg|class=skin-invert]]
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commute.

A morphism <math>f\colon (x,h)\to(x',h')</math> of <math>T</math>-algebras is an arrow <math>f\colon x\to x'</math> of <math>C</math> such that the diagram [[Image:Monad_morphism_algebra.svg|center]] commutes. <math>T</math>-algebras form a category called the ''Eilenberg–Moore category'' and denoted by <math>C^T</math>.

=== Examples ===

==== Algebras over the free group monad ====
For example, for the free group monad discussed above, a <math>T</math>-algebra is a set <math>X</math> together with a map from the free group generated by <math>X</math> towards <math>X</math> subject to associativity and unitality conditions. Such a structure is equivalent to saying that <math>X</math> is a group itself.

==== Algebras over the distribution monad ====
Another example is the '''''distribution monad''''' <math>\mathcal{D}</math> on the category of sets. It is defined by sending a set <math>X</math> to the set of functions <math>f : X \to [0,1]</math> with finite support and such that their sum is equal to <math>1</math>. In set-builder notation, this is the set<math display="block">\mathcal{D}(X) = \left\{
f: X \to [0,1] : \begin{matrix}
\#\text{supp}(f) < +\infty \\
\sum_{x \in X} f(x) = 1
\end{matrix}
\right\}</math>By inspection of the definitions, it can be shown that algebras over the distribution monad are equivalent to [[convex set]]s, i.e., sets equipped with operations <math>x +_r y</math> for <math>r \in [0,1]</math> subject to axioms resembling the behavior of convex linear combinations <math>rx + (1-r)y</math> in Euclidean space.<ref>{{Citation |last=Świrszcz |first=T. |title=Monadic functors and convexity|journal=Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys.|volume=22 |year=1974 |pages=39–42|mr=0390019}},

{{Citation|doi=10.1007/978-3-642-15240-5_1|chapter=Convexity, Duality and Effects|title=Theoretical Computer Science|volume=323 |pages=1–19|series=IFIP Advances in Information and Communication Technology|year=2010|last1=Jacobs|first1=Bart |isbn=978-3-642-15239-9 |doi-access=free}}</ref>

==== Algebras over the symmetric monad ====
Another useful example of a monad is the symmetric algebra functor on the category of <math>R</math>-modules for a commutative ring <math>R</math>.<math display="block">\text{Sym}^\bullet(-): \text{Mod}(R) \to \text{Mod}(R)</math>sending an <math>R</math>-module <math>M</math> to the direct sum of [[symmetric tensor]] powers<math display="block">\text{Sym}^\bullet(M) = \bigoplus_{k=0}^\infty \text{Sym}^k(M)</math>where <math>\text{Sym}^0(M) = R</math>. For example, <math>\text{Sym}^\bullet(R^{\oplus n}) \cong R[x_1,\ldots, x_n]</math> where the <math>R</math>-algebra on the right is considered as a module. Then, an algebra over this monad are commutative <math>R</math>-algebras. There are also algebras over the monads for the alternating tensors <math>\text{Alt}^\bullet(-)</math> and total tensor functors <math>T^\bullet(-)</math> giving anti-symmetric <math>R</math>-algebras, and free <math>R</math>-algebras, so<math display="block">\begin{align}
\text{Alt}^\bullet(R^{\oplus n}) &= R(x_1,\ldots, x_n)\\
\text{T}^\bullet(R^{\oplus n}) &= R\langle x_1,\ldots, x_n \rangle
\end{align}</math>where the first ring is the free anti-symmetric algebra over <math>R</math> in <math>n</math>-generators and the second ring is the free algebra over <math>R</math> in <math>n</math>-generators.

==== 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.