nLab
operad for bimodules

Contents

Idea

An operad whose algebras over an operad are triples consistsing of two associative algebras AA, BB and an AA-BB-bimodule .

Definition

Definition

Write BModBMod for the colored symmetric operad whose

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

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

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

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

    • if Y=𝔫Y = \mathfrak{n}: the set of linear orders {i 1<<i n}\{i_1 \lt \cdots \lt i_n\} such that there is exactly one index i ki_k with X i k=𝔫X_{i_k} = \mathfrak{n} and X i j=𝔞 X_{i_j} = \mathfrak{a}_- for all j<kj \lt k and X i j=𝔞 +X_{i_j} = \mathfrak{a}_+ for all k>kk \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 AssocBModAssoc \to BMod of the associative operad given by labelling its unique color/object with either 𝔞 \mathfrak{a}_- or 𝔞 +\mathfrak{a}_+, respectively. For

(A 1,A 2,N):BMod 𝒞 (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:Assoc BMod (A 1,A 2,N)𝒞 . A_i \colon Assoc^\otimes \to BMod^\otimes \stackrel{(A_1, A_2, N)}{\to} \mathcal{C}^\otimes \,.

There is also a morphism BModAssocBMod \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:Assoc 𝒞 A \colon Assoc^\otimes \to \mathcal{C}^\otimes

an algebra, the composite

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

exhibits AA 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

LModBMod LMod \to BMod

etc.

Relation to bitensoring

Definition

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

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

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

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

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

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

References

Section 4.3.1 in

Last revised on February 12, 2013 at 08:14:27. See the history of this page for a list of all contributions to it.