# nLab operad for modules over an algebra

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

An operad whose algebras over an operad are pairs consisting of an associative algebra and a module over that algebra

## Definition

###### Definition

The operad for modules over an algebra $LM$ is the colored symmetric operad whose

• objects are two elements, to be denoted $\mathfrak{a}$ and $\mathfrak{n}$;

• multimorphisms$(X_i)_{i = 1}^n \to Y$ form

• if $Y = \mathfrak{a}$ and $X_i = \mathfrak{a}$ for all $i$ then: the set of linear orders on $n$ elements, equivalently the elements of the symmetric group $\Sigma_n$;

• if $Y = \mathfrak{n}$ and exactly one of the $X_i = \mathfrak{n}$ then: the set of linear order $\{i_1 \lt \cdots \lt i_n\}$ such that $X_{i_n} = \mathfrak{n}$;

• otherwise: the empty set;

• composition is given by composition of linear orders as for the associative operad.

In (Lurie) this appears as def. 4.2.1.1.

## Properties

### Relation to the associative operad

Write $Assoc$ for the associative operad, regarded as a symmetric operad.

There is a canonical inclusion morphism

$Assoc \to LM$

given by labeling the unique object/color of $Assoc$ with $\mathfrak{a}$. For $(A,N) \colon LM^\otimes \to \mathcal{C}^\otimes$ a map to a symmetric monoidal category the composite

$A \colon Assoc^\otimes \to LM^\otimes \stackrel{(A,N)}{\to} \mathcal{C}^\otimes$

is the corresponding associative algebra.

There is also a morphism

$LM \to Assoc$

given by forgetting the color and just remembering the linear orders. This is such that for

$A \colon Assoc^\otimes \to \mathcal{C}^\otimes$

and algebra, the composite

$(A,A) \colon LM^{\otimes} \to Assoc^{\otimes} \to \mathcal{C}^\otimes$

is the algebra canonically regarded as a module over itself.

### Relation to the operad for bimodules

Analogous relations exist to the operad for bimodules over algebras.

(…)

Section 4.2.1 of