nLab (infinity,1)-bimodule

Idea

Definition

$(\infty,1)$-Category of $(\infty,1)$-Bimodules and intertwiners

Write $BMod^\otimes$ for the (∞,1)-category of operators of the (∞,1)-operad operad for bimodules. Write

$$\iota_{\pm} \colon Assoc \to BMod$$

for the two canonical inclusions of the associative operad (as discussed at operad for bimodules - relation to the associative operad).

Definition (Notation)

For $p \colon \mathcal{C}^\otimes \to BMod^\otimes$ a fibration of (∞,1)-operads, write

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

for the two fiber products of $p$ with the inclusions $\iota_\pm$. The canonical projection maps

$$\mathcal{C}^\otimes_{\pm} \to Assoc^\otimes$$

exhibit these as two planar (∞,1)-operads.

Finally write

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

for the (∞,1)-category over the object labeled $\mathfrak{n}$.

(Lurie, notation 4.3.1.11).

Remark

This exhibits $\mathcal{C}$ as equipped with weak tensoring over $\mathcal{C}_-$ and reverse weak tensoring over $\mathcal{C}_+$.

The most familiar special case of these definitions to keep in mind is the following.

Remark

For $\mathcal{C}^\otimes \to Assoc^\otimes$ a coCartesian fibration of (∞,1)-operads, hence exhibiting $\mathcal{C}^\otimes$ as a monoidal (∞,1)-category, pullback along the canonical map $\phi \colon BMod^\otimes \to Assoc^\otimes$ gives a fibration

$$\phi^* \mathcal{C}^\otimes \to BMod^\otimes$$

as in def. above. In the terminology there this exhibts $\mathcal{C}$ as weakly enriched (weakly tensored) over itself from the left and from the right.

This is the special case for which bimodules are traditionally considered.

(Lurie, example 4.3.1.15)

Definition

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

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

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

Composition with the two inclusions $\iota_{1,2}\colon Assoc BMod$ of the associative operad yields a fibration in the model structure for quasi-categories $BMod(\mathcal{C}) \to Alg(\mathcal{C}_-)\times Alg(\mathcal{C}_+)$. Then for $A_- \in Alg_{\mathcal{C}_-}$ and $A_+ \in Alg_{\mathcal{C}_+}$ two algebras the fiber product

$${}_A BMod_{B}(\mathcal{C}) \coloneqq \{A\} \underset{Alg(\mathcal{C}_-)}{\times} BMod(\mathcal{C}) \underset{Alg(\mathcal{C}_-)}{\times} \{B\}$$

we call the $(\infty,1)$-category of $A$-$B$-bimodules.

(Lurie, def. 4.3.1.12)

Example

For the special case of remark where the bitensored structure on $\mathcal{C}$ is induced from a monoidal structure $\mathcal{C}^\otimes \to Asoc^\otimes$, we have by the universal property of the pullback that

$$BMod(\mathcal{C}) \simeq {Alg_{BMod}}_{/Assoc}(\mathcal{C}) \simeq \left\{ \array{ && \mathcal{C} \\ &{}^{\mathllap{(A,B,N)}}\nearrow& \downarrow \\ BMod^\otimes &\to& Assoc^\otimes } \right\}$$

Remark

Let $\mathcal{C}$ be a 1-category, for simplicity. Then a morphism

$$(A_1,B_1,N_1) \to$$

in $BMod(\mathcal{C})$ is a pair $\phi_1 \colon A_1 \to A_1$, $\rho \colon B_1 \to B_2$ of algebra homomorphisms and a morphism $\kappa \colon N_1 \to N_2$ which is “linear in both $A$ and $B$” or “is an intertwiner” with respect to $\phi$ and $\rho$ in that for all $a \in A$, $b \in B$ and $n \in N$ we have

$$\kappa(a \cdot n \cdot b) = \phi(a) \cdot \kappa(n) \,.$$

It is natural to depict this by the square diagram

$$\array{ A_1 &\stackrel{N_1}{\to}& B_1 \\ {}^{\mathllap{\phi}}\downarrow & \Downarrow^{\kappa} & \downarrow^{\mathrlap{\rho}} \\ A_2 &\underset{N_2}{\to}& B_2 } \,.$$

This notation is naturally suggestive of the existence of the further horizontal composition by tensor product of (∞,1)-modules, which we come to below.

On the other hand, a morphism $N_1 \to N_2$ in ${}_A BMod(\mathcal{C})_B$ is given by the special case of the above for $\phi = id$ and $\rho = id$.

Tensor products of $(\infty,1)$-Bimodules

Definition (Notation)

Write $Tens^\otimes$ for the generalized (∞,1)-operad discussed at tensor product of ∞-modules.

For $S \to \Delta^{op}$ an (∞,1)-functor (given as a map of simplicial sets from a quasi-category $S$ to the nerve of the simplex category), write

$$Tens^\otimes_{S} \coloneqq Tens^\otimes \underset{\Delta^{op}}{\times} S$$

for the fiber product in sSet.

Moreover, for $\mathcal{C}^\otimes \to Tens^\otimes_S$ a fibration in the model structure for quasi-categories which exhibits $\mathcal{C}^\otimes$ as an $S$-family of (∞,1)-operads, write

$$Alg_S(\mathcal{C}) \hookrightarrow Fun_{Tens^\otimes_S}(Step_S, \mathcal{C}^\otimes)$$

for the full sub-(∞,1)-category on those (∞,1)-functors which send inert morphisms to inert morphisms.

(Lurie, notation 4.3.4.15)

The $(\infty,2)$-Category of $(\infty,1)$-algebras and -bimodules

We discuss the generalization of the notion of bimodules to homotopy theory, hence the generalization from category theory to (∞,1)-category theory. (Lurie, section 4.3).

Let $\mathcal{C}$ be monoidal (∞,1)-category such that

1. it admits geometric realization of simplicial objects in an (∞,1)-category (hence a left adjoint (∞,1)-functor ${\vert-\vert} \colon \mathcal{C}^{\Delta^{op}} \to \mathcal{C}$ to the constant simplicial object functor), true notably when $\mathcal{C}$ is a presentable (∞,1)-category;

2. the tensor product $\otimes \colon \mathcal{C}\times \mathcal{C} \to \mathcal{C}$ preserves this geometric realization separately in each argument.

Then there is an (∞,2)-category $Mod(\mathcal{C})$ which given as an (∞,1)-category object internal to (∞,1)Cat has

• $(\infty,1)$-category of objects

  $$Mod(\mathcal{C})_{[0]} \simeq Alg(\mathcal{C})$$

  the A-∞ algebras and ∞-algebra homomorphisms in $\mathcal{C}$;

• $(\infty,1)$-category of morphisms

  $$Mod(\mathcal{C})_{[1]} \simeq BMod(\mathcal{C})$$

  the $\infty$-bimodules and bimodule homomorphisms (intertwiners) in $\mathcal{C}$

This is (Lurie, def. 4.3.6.10, remark 4.3.6.11).

Morover, the horizontal composition of bimodules in this (∞,2)-category is indeed the relative tensor product of ∞-modules

$$\circ_{A,B,C} = (-) \otimes_B (-) \;\colon\; {}_A Mod_{B} \times {}_{B}Mod_C \to {}_A Mod_C \,.$$

This is (Lurie, lemma 4.3.6.9 (3)).

Here are some steps in the construction:

The idea of the following constructions is this: we start with a generalized (∞,1)-operad $Tens^\otimes \to FinSet_* \times \Delta^{op}$ which is such that the (∞,1)-algebras over an (∞,1)-operad over its fiber over $[k] \in \Delta^{op}$ is equivalently the collection of $(k+1)$-tuples of A-∞ algebras in $\mathcal{C}$ together with a string of $k$ $\infty$-bimodules between them. Then we turn that into a simplicial object in (∞,1)Cat

$$Mod(\mathcal{C}) \in ((\infty,1)Cat)^{\Delta^{op}} \,.$$

This turns out to be an internal (∞,1)-category object in (∞,1)Cat, hence an (∞,2)-category whose object of objects is the category $Alg(\mathcal{C})$ of A-∞ algebras and homomorphisms in $\mathcal{C}$ between them, and whose object of morphisms is the category $BMod(\mathcal{C})$ of $\infty$-bimodules and intertwiners.

Definition

Define $Mod(\mathcal{C}) \to \Delta^{op}$ as the map of simplicial sets with the universal property that for every other map of simplicial set $K \to \Delta^{op}$ there is a canonical bijection

$$Hom_{sSet/\Delta^{op}}(K, Mod(\mathcal{C})) \simeq Alg_{Tens_K / Assoc}( \mathcal{C} ) \,,$$

where

• on the left we have the hom-simplicial set in the slice category

• on the right we have the (∞,1)-category of (∞,1)-algebras over an (∞,1)-operad given by lifts $\mathcal{A}$ in

  $$\array{ && \mathcal{C}^\otimes \\ &{}^{\mathcal{A}}\nearrow& \downarrow \\ Tens_K &\to& Assoc } \,.$$

This is (Lurie, cor. 4.3.6.2) specified to the case of (Lurie, lemma 4.3.6.9). Also (Lurie, def. 4.3.4.19)

References

The general theory in terms of higher algebra of (∞,1)-operads is discussed in section 4.3 of

• Jacob Lurie, Higher Algebra

Specifically the homotopy theory of A-infinity bimodules? is discussed in

• Volodymyr Lyubashenko, Oleksandr Manzyuk, A-infinity-bimodules and Serre A-infinity-functors (arXiv:math/0701165)

and section 5.4.1 of

• Boris Tsygan, Noncommutative calculus and operads in

  Guillermo Cortinas (ed.) Topics in Noncommutative geometry, Clay Mathematics Proceedings volume 16

The generalization to (infinity,n)-modules is in

• Rune Haugseng, The higher Morita category of $E_n$-algebras (arXiv:1412.8459)