nLab tensor product of infinity-modules

Idea

The generalization of the notion of tensor product of modules to ∞-modules.

Definition

We start by defining a collection of colored symmetric operad $Tens^\otimes$ parameterized by the simplex category $\Delta$ such that for each $k$-simplex $[k] \in Delta$ the algebras over an operad over $Tens^\otimes_{[k]}$ are $(n+1)$-tuples of associative algebras $(A_i)$ together with a consecutive sequence of bimodules over these (the right algebra of every bimodule being the left algebra of the next one).

The definition is a straightforward generalization of the of the operad for modules and the operad for bimodules.

Definition

Write $Tens^\otimes$ for the category (to be thought of as a family of categories of operators of symmetric operads) whose

• objects are triples consisting of

  • an object $\langle n\rangle \in Assoc^\otimes$ of the category of operators of the associative operad;

  • an object $[k] \in \Delta$ of the simplex category;

  • two functions $c_-, c_+ \colon \langle n\rangle^\circ \to [k]$ such that for all $i in \langle n\rangle^\circ$ either $c_+(i) = c_-(i)$ or $c_+(i) = c_-(i) + 1$;

• morphisms consist if

  • a morphism $\alpha \colon \langle n\rangle \to \langle n'\rangle$ in $Assoc^\otimes$

  • a morphism $\lambda \colon [k'] \to [k]$ in $\Delta$

  such that (…)

(Lurie, def. 4.3.4.1)

We disuss how an object of this category is to be thought of as labeled with “algebra labels $\mathfrak{a}_i$” for vertices of a simplex, an “bimodule lables $\mathfrak{n}_{i, j}$” for edges of the simplex.

Remark

By construction there are forgetful functors

$$\Delta^{op} \leftarrow Tens^\otimes \rightarrow \mathcal{Ass}^\otimes \,.$$

(Lurie, 4.3.4.1)

Definition (Notation)

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.

(Lurie, notation 4.3.4.5, 4.3.4.15)

Proposition

We have

• $Tens^\otimes_{[0]} \simeq Assoc^\otimes$, the associative operad;

• $Tens^\otimes_{[1]} \simeq BM^\otimes$ the operad for bimodules.

• $Tens^\otimes_{[k]} \simeq Tens^\otimes_{\{0,1\}} \underset{Tens^\otimes_{\{1\}}}{\coprod} Tens^\otimes_{\{1,2\}} \underset{Tens^\otimes_{\{2\}}}{\coprod} \cdots \underset{Tens^\otimes_{\{k-1\}}}{\coprod} Tens^\otimes_{\{k-1,k\}}$

  as an (∞,1)-colimit in the (∞,1)-category of (∞,1)-operads (a dual Segal condition)

(Lurie, example 4.3.4.6, 4.3.4.7, prop. 4.3.4.11)

Remark

Prop. implies that for $\mathcal{C}^\otimes$ an (∞,1)-operad, the (∞,1)-algebras over an (∞,1)-operad over the fiber $Tens^\otimes_{[k]}$ in $\mathcal{C}$ form the (∞,1)-category

$$Alg_{Tens^\otimes_{[k]}}(\mathcal{C}) \simeq \underbrace{ BMod(\mathcal{C}) \underset{Alg(\mathcal{C})}{\times} BMod(\mathcal{C}) \underset{Alg(\mathcal{C})}{\times} \cdots \underset{Alg(\mathcal{C})}{\times} BMod(\mathcal{C}) }_{k\;factors} \,.$$

(Lurie, 4.3.5)

Definition (Notation)

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)

Proposition

For an (∞,1)-functor $S \to \Delta^{op}$ and a fibration in the model structure for quasicategories $q \colon \mathcal{C}^\otimes \to Tens_S^\otimes$ exhibiting $\mathcal{C}^\otimes$ as an $S$-family of (∞,1)-operads, then there is an equivalence of (∞,1)-categories

$$Alg_{/Tens_S}(\mathcal{C}) \to Alg_S(\mathcal{C}) \,.$$

(Lurie, prop. 4.3.4.17).

Definition (Notation)

Let $\Delta^1 \to \Delta^{op}$ be the map that picks the morphism $\{0,2\} \hookrightarrow \Delta^2$ in the simplex category. With def. write

$$Tens^\otimes_{\gt} \coloneqq Tens_{\Delta^1}^\otimes \coloneqq Tens^\otimes \underset{\Delta^{op}}{\times} \Delta^1 \,.$$

(Lurie, notation 4.3.5.1)

Remark

The $Tens^\otimes_{\gt}$ of def. is a correspondence of (∞,1)-operads which exhibits bilinear maps as follows:

An ∞-algebra over an (∞,1)-operad $\gamma_1 \colon Tens^\otimes_{\gt} \times_{\Delta^1} \{1\} \to \mathcal{C}^\otimes$ is equivalently a bimodule

$$X \in {}_{A'} Mod(\mathcal{C})_{C'} \,,$$

while an $\infty$-algebra $\gamma_0 \colon Tens^\otimes_{\gt} \times_{\Delta^1} \{0\} \to \mathcal{C}^\otimes$ is equivalently a pair of bimodules

$$N_1 \in {}_A Mod(\mathcal{C})_B \;\;, \;\; N_2 \in {}_B Mod(\mathcal{C})_C \,.$$

An extension of $(\gamma_0, \gamma_1)$ through the correspondence hence to a map of generalized (∞,1)-operads $Tens^\otimes_{\gt} \to \mathcal{C}^\otimes$ is equivalently a pair of A-∞ algebra maps $A \to A'$ and $B \to B'$ together with a bilinear map $N_1 \otimes N_2 \to X$.

([Lurie, beginning of 4.3.4]).

Definition

(relative tensor product of $\infty$-bimodules)

For $q \colon \mathcal{C}^\otimes \to \mathcal{O}^\otimes$ a fibration of (∞,1)-operads, consider a morphism of generalized (∞,1)-operads

$$F \colon Tens_{\gt}^\otimes \to \mathcal{C}^{\otimes} \,.$$

This exhibits three A-∞ algebras $A_i \coloneqq F|_{\{i\}}$, a pair of bimodule objects

$$(N_1, N_2) = F|_{[2]}$$

over $A_0$-$A_1$ and over $A_1$-$A_2$, respectively, and a bimodule object $N = F|_{[1]}$ over $A_0$-$A_2$. We say that $N$ exhibits the relative tensor product of ∞-modules of $N_1$ with $N_2$ over $A_1$

$$N \simeq N_1 \otimes_{A_1} N_2$$

if $F$ is an operadic $q$-(∞,1)-colimit-diagram.

(Lurie, def. 4.3.5.3).

Remark

Let $\mathcal{C}^\otimes \to Assoc^\otimes$ exhibit a monoidal (∞,1)-category such that $\mathcal{C}$ has geometric realization of simplicial objects and the tensor product preserves these separately in each argument.

Then the tensor product of $\infty$-modules def. extends to an (∞,1)-functor

$$BMod(\mathcal{C}) \underset{Alg(\mathcal{C})}{\times} BMod(\mathcal{C}) \to BMod(\mathcal{C}) \,.$$

(Lurie, example 4.3.5.11)

Related concepts

• (∞,1)-bimodules, (∞,2)-category of (∞,1)-bimodules?

References

Section 4.3.5 of

• Jacob Lurie, Higher Algebra