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

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

ι ±:AssocBMod \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:𝒞 BMod p \colon \mathcal{C}^\otimes \to BMod^\otimes a fibration of (∞,1)-operads, write

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

for the two fiber products of pp with the inclusions ι ±\iota_\pm. The canonical projection maps

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

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

Finally write

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

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

(Lurie, notation


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

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


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

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

as in def. above. In the terminology there this exhibts 𝒞\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


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

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

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

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

we call the (,1)(\infty,1)-category of AA-BB-bimodules.

(Lurie, def.


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

BMod(𝒞)Alg BMod /Assoc(𝒞){ 𝒞 (A,B,N) BMod Assoc } 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\}

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

(A 1,B 1,N 1) (A_1,B_1,N_1) \to

in BMod(𝒞)BMod(\mathcal{C}) is a pair ϕ 1:A 1A 1\phi_1 \colon A_1 \to A_1, ρ:B 1B 2\rho \colon B_1 \to B_2 of algebra homomorphisms and a morphism κ:N 1N 2\kappa \colon N_1 \to N_2 which is “linear in both AA and BB” or “is an intertwiner” with respect to ϕ\phi and ρ\rho in that for all aAa \in A, bBb \in B and nNn \in N we have

κ(anb)=ϕ(a)κ(n). \kappa(a \cdot n \cdot b) = \phi(a) \cdot \kappa(n) \,.

It is natural to depict this by the square diagram

A 1 N 1 B 1 ϕ κ ρ A 2 N 2 B 2. \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 1N 2N_1 \to N_2 in ABMod(𝒞) B{}_A BMod(\mathcal{C})_B is given by the special case of the above for ϕ=id\phi = id and ρ=id\rho = id.

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

Definition (Notation)

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

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.

Moreover, 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

The (,2)(\infty,2)-Category of (,1)(\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 𝒞\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 ||:𝒞 Δ op𝒞{\vert-\vert} \colon \mathcal{C}^{\Delta^{op}} \to \mathcal{C} to the constant simplicial object functor), true notably when 𝒞\mathcal{C} is a presentable (∞,1)-category;

  2. the tensor product :𝒞×𝒞𝒞\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(𝒞)Mod(\mathcal{C}) which given as an (∞,1)-category object internal to (∞,1)Cat has

This is (Lurie, def., remark

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

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

This is (Lurie, lemma (3)).

Here are some steps in the construction:

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

Mod(𝒞)((,1)Cat) Δ op. 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(𝒞)Alg(\mathcal{C}) of A-∞ algebras and homomorphisms in 𝒞\mathcal{C} between them, and whose object of morphisms is the category BMod(𝒞)BMod(\mathcal{C}) of \infty-bimodules and intertwiners.


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

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


This is (Lurie, cor. specified to the case of (Lurie, lemma Also (Lurie, def.


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

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

and section 5.4.1 of

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