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Idea

An operad is a structure whose elements are formal operations, closed under the operation of plugging some formal operations into others. An algebra over an operad is a structure in which the formal operations are interpreted as actual operations on an object, via a suitable action.

Accordingly, there is a notion of module over an algebra over an operad.

Definition

Let $M$ be a closed symmetric monoidal category with monoidal unit $I$, and let $X$ be any object. There is a canonical or tautological operad $Op(X)$ whose $n^{th}$ component is the internal hom $M(X^{\otimes n}, X)$; the operad identity is the map

and the operad multiplication is given by the composite

Let $O$ be any operad in $M$. An algebra over $O$ is an object $X$ equipped with an operad map $\xi: O \to Op(X)$. Alternatively, the data of an $O$-algebra is given by a sequence of maps

which specifies an action of $O$ via finitary operations on $X$, with compatibility conditions between the operad multiplication and the structure of plugging in $k$ finitary operations on $X$ into a $k$-ary operation (and compatibility with actions by permutations).

An algebra over an operad can equivalently be defined as a category over an operad which has a single object.

If $M$ is cocomplete, then an operad in $M$ may be defined as a monoid in the symmetric monoidal category $(M^{\mathbb{P}^{op}}, \circ)$ of permutation representations in $M$, aka species in $M$, with respect to the substitution product $\circ$. There is an actegory structure $M^{\mathbb{P}^{op}} \times M \to M$ which arises by restriction of the monoidal product $\circ$ if we consider $M$ as fully embedded in $M^{\mathbb{P}^{op}}$:

(interpret $X$ as concentrated in the 0-ary or “constants” component), so that an operad $O$ induces a monad $\hat{O}$ on $M$ via the actegory structure. As a functor, the monad may be defined by a coend formula

An $O$-algebra is the same thing as an algebra over the monad $\hat{O}$.

Remark If $C$ is the symmetric monoidal enriching category, $O$ the $C$-enriched operad in question, and $A \in Obj(C)$ is the single hom-object of the O-category with single object, it makes sense to write $\mathbf{B}A$ for that $O$-category. Compare the discussion at monoid and group, which are special cases of this.

Examples

Over single-coloured operads

• an associative algebra is an algebra over the associative operad.

  • an A-infinity algebra is an algebra over a cofibrant resolution of $Assoc$.

• a commutative algebra is an algebra over the commutative operad.

  • an E-infinity algebra is an algebra over a cofibrant resolution of the commutative operad

• etc.

Over coloured operads

• There is a coloured operad $Mod_P$ whose algebras are pairs consisting of a $P$-algebra $A$ and a module over $A$;

• For a single-coloured operad $P$ there is a coloured operad $P^1$ whose algebras are triples consisting of two $P$ algebras and a morphism $A_1 \to A_2$ between them.

• Let $C$ be a set. There is a $C$-coloured operad whose algebras are $V$-enriched categories with $C$ as their set of objects.

Literature

Related $n$Lab entries

• algebra over a monad

  ∞-algebra over an (∞,1)-monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

  • homotopy T-algebra / model structure on simplicial T-algebras

• algebra over an operad

  ∞-algebra over an (∞,1)-operad

  • model structure on algebras over an operad

Generalizations

• S. N. Tronin, Algebras over multicategories, Russ Math. (2016) 60: 52. doi; Rus. original: С. Н. Тронин, Об алгебрах над мультикатегориями, Изв. вузов. Матем., 2016, № 2, 62–74