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 be a closed symmetric monoidal category with monoidal unit , and let be any object. There is a canonical or tautological operad whose component is the internal hom ; the operad identity is the map
and the operad multiplication is given by the composite
Let be any operad in . An algebra over is an object equipped with an operad map . Alternatively, the data of an -algebra is given by a sequence of maps
which specifies an action of via finitary operations on , with compatibility conditions between the operad multiplication and the structure of plugging in finitary operations on into a -ary operation (and compatibility with actions by permutations).
If is cocomplete, then an operad in may be defined as a monoid in the symmetric monoidal category of permutation representations in , aka species in , with respect to the substitution product . There is an actegory structure which arises by restriction of the monoidal product if we consider as fully embedded in :
(interpret as concentrated in the 0-ary or “constants” component), so that an operad induces a monad on via the actegory structure. As a functor, the monad may be defined by a coend formula
An -algebra is the same thing as an algebra over the monad .
Remark If is the symmetric monoidal enriching category, the -enriched operad in question, and is the single hom-object of the O-category with single object, it makes sense to write for that -category. Compare the discussion at monoid and group, which are special cases of this.
For a single-coloured operad there is a coloured operad whose algebras are triples consisting of two algebras and a morphism between them.
Let be a set. There is a -coloured operad whose algebras are -enriched categories with as their set of objects.
algebra over an operad