nLab operad for bimodules

Idea

An operad whose algebras over an operad are triples consistsing of two associative algebras $A$, $B$ and an $A$-$B$-bimodule .

Definition

Definition

Write $BMod$ for the colored symmetric operad whose

• objects are three elements, to be denoted $\mathfrak{a}_-, \mathfrak{a}_+$ and $\mathfrak{n}$;

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

  • if $Y = \mathfrak{a}_-$ and all $X_i = \mathfrak{a}_-$ then: the set of linear orders of $n$ elements;

  • if $Y = \mathfrak{a}_*$ and all $X_i = \mathfrak{a}_*$ then again: the set of linear orders of $n$ elements;

  • if $Y = \mathfrak{n}$: the set of linear orders $\{i_1 \lt \cdots \lt i_n\}$ such that there is exactly one index $i_k$ with $X_{i_k} = \mathfrak{n}$ and $X_{i_j} = \mathfrak{a}_-$ for all $j \lt k$ and $X_{i_j} = \mathfrak{a}_+$ for all $j \gt k$.

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

Properties

Relation to the associative operad

There are two canonical inclusions $Assoc \to BMod$ of the associative operad given by labelling its unique color/object with either $\mathfrak{a}_-$ or $\mathfrak{a}_+$, respectively. For

$$(A_1,A_2,N) \colon BMod^\otimes \to \mathcal{C}^\otimes$$

a morphism to a symmetric monoidal category, there compositions pick the left and the right algebra

$$A_i \colon Assoc^\otimes \to BMod^\otimes \stackrel{(A_1, A_2, N)}{\to} \mathcal{C}^\otimes \,.$$

There is also a morphism $BMod \to Assoc$ given by forgetting the labels and just remembering the linear orders.

Proposition

This is a fibration of (∞,1)-operads.

In (Lurie) this appears as remark 4.3.1.8.

This is such that for

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

an algebra, the composite

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

exhibits $A$ canonically as a bimodule over itself.

Relation to the operad for modules over an algebra

Similarly, there is an inclusion of the operad for modules over an algebra

$$LMod \to BMod$$

etc.

Relation to bitensoring

Definition

A coCartesian fibration of (∞,1)-operads $\mathcal{C}^\otimes \to BMod^\otimes$, hence the structure of a $BMod$-monoidal (∞,1)-category is a bitensoring of

$$\mathcal{C} \coloneqq \mathcal{C}^\otimes \underset{BMod}{\times} \{\mathfrak{n}\}$$

over the (ordinary) monoidal (∞,1)-categories

$$\mathcal{C}^\otimes_{\pm} \coloneqq \mathcal{C}^\otimes \underset{BMod}{\times} Assoc^\otimes_{\pm} \,.$$

(Lurie, def. 4.3.1.17)

Remark

By the microcosm principle, bitensored $(\infty,1)$-categories are the right context into which to internalize bimodules. See Relation to the category of bimodules below.

Relation to categories of bimodules

For $\mathcal{C}^\otimes \to BMod^\otimes$ a fibration of (∞,1)-operads the corresponding (∞,1)-category of (∞,1)-algebras over an (∞,1)-operad

$$BMod(\mathcal{C}) \coloneqq Alg_{/BMod}(\mathcal{C})$$

is the (∞,1)-category of (∞,1)-bimodules in $\mathcal{C}$.

Related concepts

• operad for modules over an algebra

References

Section 4.3.1 in

• Jacob Lurie, Higher Algebra