nLab
operad for modules over an algebra

Contents

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 LMLM is the colored symmetric operad whose

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

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

    • if Y=𝔞Y = \mathfrak{a} and X i=𝔞X_i = \mathfrak{a} for all ii then: the set of linear orders on nn elements, equivalently the elements of the symmetric group Σ n\Sigma_n;

    • if Y=𝔫Y = \mathfrak{n} and exactly one of the X i=𝔫X_i = \mathfrak{n} then: the set of linear order {i 1<<i n}\{i_1 \lt \cdots \lt i_n\} such that X i n=𝔫X_{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 AssocAssoc for the associative operad, regarded as a symmetric operad.

There is a canonical inclusion morphism

AssocLM Assoc \to LM

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

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

is the corresponding associative algebra.

There is also a morphism

LMAssoc LM \to Assoc

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

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

and algebra, the composite

(A,A):LM Assoc 𝒞 (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.

(…)

References

Section 4.2.1 of

Last revised on February 12, 2013 at 10:03:43. See the history of this page for a list of all contributions to it.