nLab operad for modules over an algebra

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.

(…)

Related concepts

• associative operad, commutative operad

• operad for bimodules over algebras

• model structure on modules over an algebra over an operad

References

Section 4.2.1 of

• Jacob Lurie, Higher Algebra