Algebra object - Knowlpedia

Definition

Let $(\mathcal C,\otimes,\mathbb 1)$ be a monoidal category. An algebra object in $\mathcal C$ is an object $A$ with multiplication and unit morphisms

\mu:A\otimes A\longrightarrow A,\qquad \eta:\mathbb 1\longrightarrow A

for which multiplication is associative, after inserting the associator, and $\eta$ is a left and right unit, after inserting the unitors. A morphism of algebra objects is a morphism in $\mathcal C$ that preserves $\mu$ and $\eta$ .

Commutative algebra objects

If $\mathcal C$ is symmetric monoidal with symmetry $\beta$ , the algebra object is commutative when

\mu\circ\beta_{A,A}=\mu.

This formulation makes the meaning of commutativity depend on the ambient symmetry. It produces ordinary commutative algebras in modules and supercommutative algebras in super vector spaces.

Examples

In the monoidal category of modules over a commutative ring $R$ , algebra objects are precisely unital associative $R$ -algebras. In a category with finite Cartesian products, algebra objects are internal monoids.

References

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, American Mathematical Society, 2015. DOI record. Relevant: §7.8.