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