Editing Monad (category theory) (section)

==== 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.