algebra over an operad



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

1 X:IM(X,X)1_X: I \to M(X, X)

and the operad multiplication is given by the composite

M(X k,X)M(X n 1,X)M(X n k,X) 1func M(X k,X)M(X n 1++n k,X k) comp M(X n 1++n k,X)\array{ M(X^{\otimes k}, X) \otimes M(X^{\otimes n_1}, X) \otimes \ldots \otimes M(X^{\otimes n_k}, X) & \stackrel{1 \otimes func_\otimes}{\to} & M(X^{\otimes k}, X) \otimes M(X^{\otimes n_1 + \ldots + n_k}, X^{\otimes k}) \\ & \stackrel{comp}{\to} & M(X^{\otimes n_1 + \ldots + n_k}, X) }

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

O(k)X kXO(k) \otimes X^{\otimes k} \to X

which specifies an action of OO via finitary operations on XX, with compatibility conditions between the operad multiplication and the structure of plugging in kk finitary operations on XX into a kk-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 MM is cocomplete, then an operad in MM may be defined as a monoid in the symmetric monoidal category (M op,)(M^{\mathbb{P}^{op}}, \circ) of permutation representations in MM, aka species in MM, with respect to the substitution product \circ. There is an actegory structure M op×MMM^{\mathbb{P}^{op}} \times M \to M which arises by restriction of the monoidal product \circ if we consider MM as fully embedded in M opM^{\mathbb{P}^{op}}:

i:MM op:X(nδ n0X)i: M \to M^{\mathbb{P}^{op}}: X \mapsto (n \mapsto \delta_{n 0} \cdot X)

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

O^(X)= kO(k)X k\hat{O}(X) = \int^{k \in \mathbb{P}} O(k) \otimes X^{\otimes k}

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

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


Over single-coloured operads

Over coloured operads

  • There is a coloured operad Mod PMod_P whose algebras are pairs consisting of a PP-algebra AA and a module over AA;

  • For a single-coloured operad PP there is a coloured operad P 1P^1 whose algebras are triples consisting of two PP algebras and a morphism A 1A 2A_1 \to A_2 between them.

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



  • 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

Revised on November 29, 2016 10:38:29 by Zoran Škoda (