symmetric monoidal (∞,1)-category of spectra
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.
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.
an associative algebra is an algebra over the associative operad.
a commutative algebra is an algebra over the commutative operad.
etc.
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.
algebra over an operad