symmetric monoidal (∞,1)-category of spectra
Write 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 a fibration of (∞,1)-operads, write
for the two fiber products of with the inclusions . The canonical projection maps
exhibit these as two planar (∞,1)-operads.
Finally write
for the (∞,1)-category over the object labeled .
This exhibits as equipped with weak tensoring over and reverse weak tensoring over .
The most familiar special case of these definitions to keep in mind is the following.
For a coCartesian fibration of (∞,1)-operads, hence exhibiting as a monoidal (∞,1)-category, pullback along the canonical map gives a fibration
as in def. above. In the terminology there this exhibts 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 a fibration of (∞,1)-operads we say that the corresponding (∞,1)-category of (∞,1)-algebras over an (∞,1)-operad
is the -category of -bimodules in .
Composition with the two inclusions of the associative operad yields a fibration in the model structure for quasi-categories . Then for and two algebras the fiber product
we call the -category of --bimodules.
For the special case of remark where the bitensored structure on is induced from a monoidal structure , we have by the universal property of the pullback that
Let be a 1-category, for simplicity. Then a morphism
in is a pair , of algebra homomorphisms and a morphism which is “linear in both and ” or “is an intertwiner” with respect to and in that for all , and 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 in is given by the special case of the above for and .
Write for the generalized (∞,1)-operad discussed at tensor product of ∞-modules.
For an (∞,1)-functor (given as a map of simplicial sets from a quasi-category to the nerve of the simplex category), write
for the fiber product in sSet.
Moreover, for a fibration in the model structure for quasi-categories which exhibits as an -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 be monoidal (∞,1)-category such that
it admits geometric realization of simplicial objects in an (∞,1)-category (hence a left adjoint (∞,1)-functor to the constant simplicial object functor), true notably when is a presentable (∞,1)-category;
the tensor product preserves this geometric realization separately in each argument.
Then there is an (∞,2)-category which given as an (∞,1)-category object internal to (∞,1)Cat has
-category of objects
the A-∞ algebras and ∞-algebra homomorphisms in ;
-category of morphisms
the -bimodules and bimodule homomorphisms (intertwiners) in
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 which is such that the (∞,1)-algebras over an (∞,1)-operad over its fiber over is equivalently the collection of -tuples of A-∞ algebras in together with a string of -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 of A-∞ algebras and homomorphisms in between them, and whose object of morphisms is the category of -bimodules and intertwiners.
Define as the map of simplicial sets with the universal property that for every other map of simplicial set 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 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
Last revised on August 1, 2015 at 10:07:54. See the history of this page for a list of all contributions to it.