symmetric monoidal (∞,1)-category of spectra
Write $BMod^\otimes$ for the (∞,1)-category of operators of the (∞,1)-operad operad for bimodules. Write
for the two canonical inclusions of the associative operad (as discussed at operad for bimodules - relation to the associative operad).
For $p \colon \mathcal{C}^\otimes \to BMod^\otimes$ a fibration of (∞,1)-operads, write
for the two fiber products of $p$ with the inclusions $\iota_\pm$. The canonical projection maps
exhibit these as two planar (∞,1)-operads.
Finally write
for the (∞,1)-category over the object labeled $\mathfrak{n}$.
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.
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
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.
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
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
we call the $(\infty,1)$-category of $A$-$B$-bimodules.
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
Let $\mathcal{C}$ be a 1-category, for simplicity. Then a morphism
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
It is natural to depict this by the square diagram
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 $\rho = id$.
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
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
for the full sub-(∞,1)-category on those (∞,1)-functors which send inert morphisms to inert morphisms.
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
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;
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
the A-∞ algebras and ∞-algebra homomorphisms in $\mathcal{C}$;
$(\infty,1)$-category of morphisms
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
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
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.
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
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
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)
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
Guillermo Cortinas (ed.) Topics in Noncommutative geometry, Clay Mathematics Proceedings volume 16
The generalization to (infinity,n)-modules is in