tensor product of infinity-modules



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


We start by defining a collection of colored symmetric operad Tens Tens^\otimes parameterized by the simplex category Δ\Delta such that for each kk-simplex [k]Delta[k] \in Delta the algebras over an operad over Tens [k] Tens^\otimes_{[k]} are (n+1)(n+1)-tuples of associative algebras (A i)(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.


Write Tens 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 nAssoc \langle n\rangle \in Assoc^\otimes of the category of operators of the associative operad;

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

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

  • morphisms consist if

    • a morphism α:nn\alpha \colon \langle n\rangle \to \langle n'\rangle in Assoc Assoc^\otimes

    • a morphism λ:[k][k]\lambda \colon [k'] \to [k] in Δ\Delta

    such that (…)

(Lurie, def.

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


By construction there are forgetful functors

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


Definition (Notation)

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

Tens S Tens ×Δ opS Tens^\otimes_{S} \coloneqq Tens^\otimes \underset{\Delta^{op}}{\times} S

for the fiber product in sSet.

(Lurie, notation,


We have

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

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

  • Tens [k] Tens {0,1} Tens {1} Tens {1,2} Tens {2} Tens {k1} Tens {k1,k} 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,, prop.


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

Alg Tens [k] (𝒞)BMod(𝒞)×Alg(𝒞)BMod(𝒞)×Alg(𝒞)×Alg(𝒞)BMod(𝒞) kfactors. 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 𝒞 Tens S \mathcal{C}^\otimes \to Tens^\otimes_S a fibration in the model structure for quasi-categories which exhibits 𝒞 \mathcal{C}^\otimes as an SS-family of (∞,1)-operads, write

Alg S(𝒞)Fun Tens S (Step S,𝒞 ) 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


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

Alg /Tens S(𝒞)Alg S(𝒞). Alg_{/Tens_S}(\mathcal{C}) \to Alg_S(\mathcal{C}) \,.

(Lurie, prop.

Definition (Notation)

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

Tens > Tens Δ 1 Tens ×Δ opΔ 1. Tens^\otimes_{\gt} \coloneqq Tens_{\Delta^1}^\otimes \coloneqq Tens^\otimes \underset{\Delta^{op}}{\times} \Delta^1 \,.

(Lurie, notation


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

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

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

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

N 1 AMod(𝒞) B,N 2 BMod(𝒞) C. N_1 \in {}_A Mod(\mathcal{C})_B \;\;, \;\; N_2 \in {}_B Mod(\mathcal{C})_C \,.

An extension of (γ 0,γ 1)(\gamma_0, \gamma_1) through the correspondence hence to a map of generalized (∞,1)-operads Tens > 𝒞 Tens^\otimes_{\gt} \to \mathcal{C}^\otimes is equivalently a pair of A-∞ algebra maps AAA \to A' and BBB \to B' together with a bilinear map N 1N 2XN_1 \otimes N_2 \to X.

([Lurie, beginning of 4.3.4]).


(relative tensor product of \infty-bimodules)

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

F:Tens > 𝒞 . F \colon Tens_{\gt}^\otimes \to \mathcal{C}^{\otimes} \,.

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

(N 1,N 2)=F| [2] (N_1, N_2) = F|_{[2]}

over A 0A_0-A 1A_1 and over A 1A_1-A 2A_2, respectively, and a bimodule object N=F| [1]N = F|_{[1]} over A 0A_0-A 2A_2. We say that NN exhibits the relative tensor product of ∞-modules of N 1N_1 with N 2N_2 over A 1A_1

NN 1 A 1N 2 N \simeq N_1 \otimes_{A_1} N_2

if FF is an operadic qq-(∞,1)-colimit-diagram.

(Lurie, def.


Let 𝒞 Assoc \mathcal{C}^\otimes \to Assoc^\otimes exhibit a monoidal (∞,1)-category such that 𝒞\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(𝒞)×Alg(𝒞)BMod(𝒞)BMod(𝒞). BMod(\mathcal{C}) \underset{Alg(\mathcal{C})}{\times} BMod(\mathcal{C}) \to BMod(\mathcal{C}) \,.

(Lurie, example


Section 4.3.5 of

Last revised on February 12, 2013 at 13:03:25. See the history of this page for a list of all contributions to it.