nLab
algebra over an operad

higher algebra

Ingredients

Higher algebraic theories

Higher algebras

monoidal (∞,1)-category
symmetric monoidal (∞,1)-category of spectra
A-∞ ring
E-∞ ring

Models for higher algebras

Edit this sidebar

Idea
Definition
Examples

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.

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

1_{X} : I \to M (X, X)

and the operad multiplication is given by the composite

\begin{matrix} M (X^{\otimes k}, X) \otimes M (X^{\otimes n_{1}}, X) \otimes \dots \otimes M (X^{\otimes n_{k}}, X) & \overset{1 \otimes {func}_{\otimes}}{\to} & M (X^{\otimes k}, X) \otimes M (X^{\otimes n_{1} + \dots + n_{k}}, X^{\otimes k}) \\ \overset{comp}{\to} & M (X^{\otimes n_{1} + \dots + n_{k}}, X) \end{matrix}

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

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

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 $(M^{ℙ^{op}}, \circ)$ of permutation representations in $M$ , aka species in $M$ , with respect to the substitution product $\circ$ . There is an actegory structure $M^{ℙ^{op}} \times M \to M$ which arises by restriction of the monoidal product $\circ$ if we consider $M$ as fully embedded in $M^{ℙ^{op}}$ :

i : M \to M^{ℙ^{op}} : X \mapsto (n \mapsto δ_{n 0} \cdot X)

(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

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

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 $B A$ for that $O$ -category. Compare the discussion at monoid and group, which are special cases of this.

Examples

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.

Revised on July 26, 2010 11:37:03 by Urs Schreiber (134.100.32.207)

nLab algebra over an operad