# nLab algebra over an operad

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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

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

and the operad multiplication is given by the composite

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

Let $O$ 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

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

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 $(M^{\mathbb{P}^{op}}, \circ)$ 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 $M^{\mathbb{P}^{op}}$:

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

(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

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

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

## Examples

• 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

### 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

Last revised on November 29, 2016 at 10:38:29. See the history of this page for a list of all contributions to it.