symmetric monoidal (∞,1)-category of spectra
A bimodule is a module in two compatible ways over two rings.
Given two rings and , a --bimodule is an abelian group with a bilinear left -action and a bilinear right -action such that for all , , and , .
Equivalently, given two rings and , a --bimodule is an abelian group with a trilinear --biaction, a function such that
for all ,
for all , , , , ,
for all , , , ,
for all , , , ,
for all , , , ,
representing simultaneous left multiplication by scalars and right multiplication by scalars .
We can define in more generality what is a -bimodule in a monoidal category where and are two monoids. It is given by:
such that:
and moreover this diagram commutes:
Let and be rings, and let be a --bimodule.
Given a left -action and a right -action of a --bimodule, the biaction is defined as
The biaction is trilinear because the left -action and right -action are bilinear.
On the other hand, given an --biaction of a --bimodule, the left -action is defined from the --biaction as
for all and . It is a left action because
The right -action is defined from the --biaction as
for all and . It is a right action because
The left -action and right -action satisfy the following identity:
This is because when expanded out, the identity becomes:
The left -action and right -action are bilinear because the original biaction is trilinear.
Let and be rings. A --linear map or --bimodule homomorphism between two --bimodules and is an abelian group homomorphism such that for all , , and ,
A --linear map is monic or an --bimodule monomorphism if for every other --bimodule and --linear maps and , implies that .
A sub---bimodule of a --bimodule is a --bimodule with a monic linear map .
A --linear map is invertible or an --bimodule isomorphism if there exists a --linear map such that and , where and are the identity linear maps on and respectively.
Given rings and an - bimodule and an - bimodule , the tensor product of and is formed as a quotient of the tensor product of abelian groups . This is a special case of a more general construction:
Given three monoids in a monoidal category , a --bimodules and a --bimodule , we denote the monoid actions as , , and . The tensor product, is defined as this coequalizer:
We suppose moreover that this coequalizer is preserved by tensoring on the left by and tensoring on the right by , meaning that these diagrams are coequalizer diagrams: becomes a --bimodule with left action defined by the following diagram: and right action defined by the following diagram:
Assuming all requisite (reflective) coequalizers exist, universal property arguments guarantee associativity isomorphisms of type
In fact, this tensor product defines composition in a bicategory where objects or 0-cells are monoids in a monoidal category, where 1-cells from to are - bimodules, and where 2-cells from to are morphisms of - bimodules.
This in turn can be seen as a special case of a bicategory of profunctors enriched in a monoidal category with suitably nice cocompleteness properties – see monoidally cocomplete category and Benabou cosmos.
Every ring is a --bimodule, with the biaction defined by the ternary product for elements , , .
Given a ring , a two-sided ideal of is a sub---bimodule of .
Let be a ring. An -ring is a --bimodule with a bilinear function and an element such that forms a monoid.
Write for the category whose
objects are triples where and are rings and where is an --bimodule;
morphisms are triples consisting of two ring homomorphisms and and an intertwiner of --bimodules . This we may depict as a
As this notation suggests, is naturally the vertical category of a pseudo double category whose horizontal composition is given by tensor product of bimodules.
Consider bimodules over rings.
There is a 2-category whose
1-morphisms are bimodules;
2-morphisms are intertwiners.
The composition of 1-morphisms is given by the tensor product of modules over the middle algebra.
There is a 2-functor from the above 2-category of rings and bimodules to Cat which
sends an ring to its category of modules ;
sends a --bimodule to the tensor product functor
sends an intertwiner to the evident natural transformation of the above functors.
This construction has as its image precisely the colimit-preserving functors between categories of modules.
This is the Eilenberg-Watts theorem.
In the context of higher category theory/higher algebra one may interpret this as says that the 2-category of those 2-modules over the given ring which are equivalent to a category of modules is that of rings, bimodules and intertwiners. See also at 2-ring.
The 2-category of rings and bimodules is an archtypical example for a 2-category with proarrow equipment, hence for a pseudo double category with niche-fillers. Or in the language of internal (infinity,1)-category-theory: it naturally induces the structure of a simplicial object in the (2,1)-category
which satisfies the Segal conditions. Here
is the category of rings and homomorphisms between them, while
is the category of def. , whose objects are pairs consisting of two rings and and an - bimodule between them, and whose morphisms are pairs consisting of two ring homomorphisms and and an intertwiner .
The above has a generalization to (infinity,1)-bimodules. See there for more.
The 2-category of bimodules in its incarnation as a 2-category with proarrow equipment appears as example 2.3 in
Bimodules in homotopy theory/higher algebra are discussed in section 4.3 of
For more on that see at (∞,1)-bimodule.
Last revised on August 20, 2024 at 13:10:19. See the history of this page for a list of all contributions to it.