symmetric monoidal (∞,1)-category of spectra
A bimodule is a module in two compatible ways over two algebras.
Let $V$ be a closed monoidal category. Recall that for $C$ a category enriched over $V$, a $C$-module is a $V$-functor $\rho : C \to V$. We think of the objects $\rho(a)$ for $a \in Obj(C)$ as the objects on which $C$ acts, and of $\rho(C(a,b))$ as the action of $C$ on these objects.
In this language a $C$-$D$ bimodule for $V$-categories $C$ and $D$ is a $V$-functor
Such a functor is also called a profunctor or distributor.
Some points are in order. Strictly speaking, the construction of $C^{op}$ from a $V$-category $C$ requires that $V$ be symmetric (or at least braided) monoidal. It’s possible to define $C$-$D$ bimodules without recourse to $C^{op}$, but then either that should be spelled out, or one should include a symmetry. (If the former is chosen, then closedness one on side might not be the best choice of assumption, in view of the next remark; a more natural choice might be biclosed monoidal.)
Second: bimodules are not that much good unless you can compose them; for that one should add some cocompleteness assumptions to $V$ (with $\otimes$ cocontinuous in both arguments; biclosedness would ensure that), and consider smallness assumptions on the objects $C$, $D$, etc. —Todd.
Let $V = Set$ and let $C = D$. Then the hom functor $C(-, -):C^{op} \times C \to Set$ is a bimodule. Bimodules can be thought of as a kind of generalized hom, giving a set of morphisms (or object of $V$) between an object of $C$ and an object of $D$.
Let $\hat{C} = Set^{C^{op}}$; the objects of $\hat{C}$ are “generating functions” that assign to each object of $C$ a set. Every bimodule $f:D^op \times C \to Set$ can be curried to give a Kleisli arrow $\tilde{f}:C \to \hat{D}$. Composition of these arrows corresponds to convolution of the generating functions.
Todd: I am not sure what is trying to be said with regard to “convolution”. I know about Day convolution, but this is not the same thing.
Also, with regard to “Kleisli arrow”: I understand the intent, but one should proceed with caution since there is no global monad $C \mapsto \hat{C}$ to which Kleisli would refer. Again there are size issues that need attending to.
Let $V = Vect$ and let $C = \mathbf{B}A_1$ and $D = \mathbf{B}A_2$ be two one-object $Vect$-enriched categories, whose endomorphism vector spaces are hence algebras. Then a $C$-$D$ bimodule is a vector space $V$ with an action of $A_1$ on the left and and action of $A_2$ on the right.
For $R$ a commutative ring, write $BMod_R$ for the category whose
objects are triples $(A,B,N)$ where $A$ and $B$ are $R$-algebras and where $N$ is an $A$-$B$-bimodule;
morphisms are triples $(f,g, \phi)$ consisting of two algebra homomorphisms $f \colon A \to A'$ and $B \colon B \to B'$ and an intertwiner of $A$-$B'$-bimdules $\phi \colon N \cdot g \to f \cdot N'$. This we may depict as a
As this notation suggests, $BMod_R$ is naturally the vertical category of a pseudo double category whose horizontal composition is given by tensor product of bimodules. spring
Let $R$ be a commutative ring and consider bimodules over $R$-algebras.
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 algebras and bimodules to Cat which
sends an $R$-algebra $A$ to its category of modules $Mod_A$;
sends a $A_1$-$A_2$-bimodule $N$ 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 $R$-algebras, bimodules and intertwiners. See also at 2-ring.
The 2-category of algebras 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 $Cat$
which satisfies the Segal conditions. Here
is the category of associative algebras and homomorphisms between them, while
is the category of def. 1, whose objects are pairs consisting of two algebras $A$ and $B$ and an $A$-$B$ bimodule $N$ between them, and whose morphisms are pairs consisting of two algebra homomorphisms $f \colon A \to A'$ and $g \colon B \to B'$ and an intertwiner $N \cdot (g) \to (f) \cdot N'$.
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.