# 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 $\mathrm{Op}\left(X\right)$ whose ${n}^{\mathrm{th}}$ component is the internal hom $M\left({X}^{\otimes n},X\right)$; the operad identity is the map

${1}_{X}:I\to M\left(X,X\right)$1_X: I \to M(X, X)

and the operad multiplication is given by the composite

$\begin{array}{ccc}M\left({X}^{\otimes k},X\right)\otimes M\left({X}^{\otimes {n}_{1}},X\right)\otimes \dots \otimes M\left({X}^{\otimes {n}_{k}},X\right)& \stackrel{1\otimes {\mathrm{func}}_{\otimes }}{\to }& M\left({X}^{\otimes k},X\right)\otimes M\left({X}^{\otimes {n}_{1}+\dots +{n}_{k}},{X}^{\otimes k}\right)\\ & \stackrel{\mathrm{comp}}{\to }& M\left({X}^{\otimes {n}_{1}+\dots +{n}_{k}},X\right)\end{array}$\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 $\xi :O\to \mathrm{Op}\left(X\right)$. Alternatively, the data of an $O$-algebra is given by a sequence of maps

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

$i:M\to {M}^{{ℙ}^{\mathrm{op}}}:X↦\left(n↦{\delta }_{n0}\cdot X\right)$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 $\stackrel{^}{O}$ on $M$ via the actegory structure. As a functor, the monad may be defined by a coend formula

$\stackrel{^}{O}\left(X\right)={\int }^{k\in ℙ}O\left(k\right)\otimes {X}^{\otimes k}$\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 $\stackrel{^}{O}$.

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

## Examples

• There is a coloured operad ${\mathrm{Mod}}_{P}$ 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.