symmetric monoidal (∞,1)-category of spectra
A bimodule is a module in two compatible ways over two rings.
Given two rings $R$ and $S$, a $R$-$S$-bimodule is an abelian group $B$ with a bilinear left $R$-action $\alpha_R:R \times B \to B$ and a bilinear right $S$-action $\alpha_S:B \times S \to B$ such that for all $r \in R$, $b \in B$, and $s \in S$, $\alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)$.
Equivalently, given two rings $R$ and $S$, a $R$-$S$-bimodule is an abelian group $B$ with a trilinear $R$-$S$-biaction, a function $(-)(-)(-):R \times B \times S \to B$ such that
for all $b \in B$, $1_R b 1_S = b$
for all $b \in B$, $r_1 \in R$, $r_2 \in R$, $s_1 \in S$, $s_2 \in S$, $r_1 (r_2 b s_1) s_2 = (r_1 \cdot_R r_2) b (s_1 \cdot_S s_2)$
for all $r_1 \in R$, $r_2 \in R$, $b \in B$, $s \in S$, $(r_1 + r_2) b s = r_1 b s + r_2 b s$
for all $r \in R$, $b_1 \in B$, $b_2 \in B$, $s \in S$, $r (b_1 + b_2) s = r b_1 s + r b_2 s$
for all $r \in R$, $b \in B$, $s_1 \in S$, $s_2 \in S$, $r b (s_1 + s_2) = r b s_1 + r b s_2$
representing simultaneous left multiplication by scalars $r \in R$ and right multiplication by scalars $s \in S$.
We can define in more generality what is a $(A,B)$-bimodule in a monoidal category $(\mathcal{C},\otimes,I)$ where $(A,\nabla^{A},\eta^{A})$ and $(B,\nabla^{B},\eta^{B})$ are two monoids. It is given by:
such that:
and moreover this diagram commutes:
Let $R$ and $S$ be rings, and let $B$ be a $R$-$S$-bimodule.
Given a left $R$-action $\alpha_R$ and a right $S$-action $\alpha_S$ of a $R$-$S$-bimodule, the biaction $(-)(-)(-):R \times B \times S \to B$ is defined as
The biaction is trilinear because the left $R$-action and right $S$-action are bilinear.
On the other hand, given an $R$-$S$-biaction $\alpha$ of a $R$-$S$-bimodule, the left $R$-action is defined from the $R$-$S$-biaction as
for all $r \in R$ and $b \in B$. It is a left action because
The right $S$-action is defined from the $R$-$S$-biaction as
for all $s \in S$ and $b \in B$. It is a right action because
The left $R$-action and right $S$-action satisfy the following identity:
This is because when expanded out, the identity becomes:
The left $R$-action and right $S$-action are bilinear because the original biaction is trilinear.
Let $R$ and $S$ be rings. A $R$-$S$-linear map or $R$-$S$-bimodule homomorphism between two $R$-$S$-bimodules $A$ and $B$ is an abelian group homomorphism $f:A \to B$ such that for all $a \in A$, $r \in R$, and $s \in S$,
A $R$-$S$-linear map $f:A \to B$ is monic or an $R$-$S$-bimodule monomorphism if for every other $R$-$S$-bimodule $C$ and $R$-$S$-linear maps $h:C \to A$ and $k:C \to A$, $f \circ h = f \circ k$ implies that $h = k$.
A sub-$R$-$S$-bimodule of a $R$-$S$-bimodule $B$ is a $R$-$S$-bimodule $A$ with a monic linear map $i:A \hookrightarrow B$.
A $R$-$S$-linear map $f:A \to B$ is invertible or an $R$-$S$-bimodule isomorphism if there exists a $R$-$S$-linear map $g:B \to A$ such that $g \circ f = id_A$ and $f \circ g = id_B$, where $id_A$ and $id_B$ are the identity linear maps on $A$ and $B$ respectively.
Given rings $R, S, T$ and an $R$-$S$ bimodule $A$ and an $S$-$T$ bimodule $B$, the tensor product of $A$ and $B$ is formed as a quotient $A \otimes_N B$ of the tensor product of abelian groups $A\otimes B$. This is a special case of a more general construction:
Given three monoids $M,N,P$ in a monoidal category $(\mathcal{C},\otimes,I)$, a $M$-$N$-bimodules $A$ and a $N$-$P$-bimodule $B$, we denote the monoid actions as $\lambda^{A}:M \otimes A \rightarrow A$, $\rho^{A}:A \otimes N \rightarrow N$, $\lambda^{B}:N \otimes B \rightarrow N$ and $\rho^{B}:B \otimes P \rightarrow P$. The tensor product, $A \otimes_{N} B$ is defined as this coequalizer:
We suppose moreover that this coequalizer is preserved by tensoring on the left by $M$ and tensoring on the right by $P$, meaning that these diagrams are coequalizer diagrams: $A \otimes B$ becomes a $M$-$P$-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 $A$ from $R$ to $S$ are $R$-$S$ bimodules, and where 2-cells from $A$ to $B$ are morphisms of $R$-$S$ 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 $R$ is a $R$-$R$-bimodule, with the biaction $(-)(-)(-):R \times R \times R \to R$ defined by the ternary product $a b c \coloneqq a \cdot b \cdot c$ for elements $a \in R$, $b \in R$, $c \in R$.
Given a ring $R$, a two-sided ideal of $R$ is a sub-$R$-$R$-bimodule of $R$.
Let $R$ be a ring. An $R$-ring $S$ is a $R$-$R$-bimodule with a bilinear function $(-)\cdot(-):S \times S \to S$ and an element $1 \in S$ such that $(S, \cdot, 1)$ forms a monoid.
Write $BMod$ for the category whose
objects are triples $(R,S,B)$ where $R$ and $S$ are rings and where $B$ is an $R$-$S$-bimodule;
morphisms are triples $(f,g, \phi)$ consisting of two ring homomorphisms $f \colon R \to R'$ and $g \colon S \to S'$ and an intertwiner of $R$-$S'$-bimodules $\phi \colon B \cdot g \to f \cdot B'$. This we may depict as a
As this notation suggests, $BMod$ 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 $R$ to its category of modules $Mod_R$;
sends a $R$-$S$-bimodule $B$ 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 $Cat$
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 $A$ and $B$ and an $A$-$B$ bimodule $N$ between them, and whose morphisms are pairs consisting of two ring 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.
Last revised on May 28, 2023 at 12:45:41. See the history of this page for a list of all contributions to it.