Double Category of Bimodules
Let
𝒱
,
⊗
,
I
𝒱
tensor-product
𝐼
\mathcal{V},\otimes,I
be a Monoidal
Category with Coequalisers . The
Double
Category
Bimod
(
𝒱
)
Bimod
𝒱
\mathrm{Bimod}(\mathcal{V})
consists of
Objects are Monoid Objects
in
𝒱
𝒱
\mathcal{V}
Tight morphisms are Homomorphisms
of Monoid Objects
Loose morphisms are Bimodule
over a Monoid Object
Composition of an
(
A
,
B
)
𝐴
𝐵
(A,B)
bimodule
M
𝑀
M
with a
(
B
,
C
)
𝐵
𝐶
(B,C)
bimodule
N
𝑁
N
is given by the Coequaliser
M ⊗ B ⊗ N tensor-product 𝑀 𝐵 𝑁 {M\otimes B\otimes N} M ⊗ N tensor-product 𝑀 𝑁 {M\otimes N} M ⊗ B N subscript tensor-product 𝐵 𝑀 𝑁 {M\otimes_{B}N} r ⊗ m a t h r m i d i f x . . ⋄ l s e i \scriptstyle{r\otimes mathrm{id}{ifx..lse\par i}} m a t h r m i d i f x . . ⋄ l s e i ⊗ l \scriptstyle{mathrm{id}{ifx..lse\par i}\otimes l}
Squares \[
\begin{tikzcd}
{A} && {B}Semilinear Maps ;
EG, morphisms
α
:
M
→
N
:
𝛼
→
𝑀
𝑁
\alpha:M\to N
that satisfy
A ⊗ M tensor-product 𝐴 𝑀 {{A\otimes M}} M 𝑀 {{M}} C ⊗ N tensor-product 𝐶 𝑁 {{C\otimes N}} N 𝑁 {{N}} l M subscript 𝑙 𝑀 \scriptstyle{l_{M}} f ⊗ α tensor-product 𝑓 𝛼 \scriptstyle{f\otimes\alpha} α 𝛼 \scriptstyle{\alpha} l N subscript 𝑙 𝑁 \scriptstyle{l_{N}} M ⊗ B tensor-product 𝑀 𝐵 {{M\otimes B}} M 𝑀 {{M}} N ⊗ D tensor-product 𝑁 𝐷 {{N\otimes D}} N 𝑁 {{N}} r M subscript 𝑟 𝑀 \scriptstyle{r_{M}} α ⊗ g tensor-product 𝛼 𝑔 \scriptstyle{\alpha\otimes g} α 𝛼 \scriptstyle{\alpha} r N subscript 𝑟 𝑁 \scriptstyle{r_{N}}
A && B && C
arise from the universal property of Coequalisers .
First, we must give a map
M
⊗
B
P
→
N
⊗
Y
Q
→
subscript
tensor-product
𝐵
𝑀
𝑃
subscript
tensor-product
𝑌
𝑁
𝑄
M\otimes_{B}P\to N\otimes_{Y}Q
; by the universal property of coequalisers, it suffices to
give a map
γ
:
M
⊗
P
→
N
⊗
Y
Q
:
𝛾
→
tensor-product
𝑀
𝑃
subscript
tensor-product
𝑌
𝑁
𝑄
\gamma:M\otimes P\to N\otimes_{Y}Q
with
γ
∘
r
M
⊗
m
a
t
h
r
m
i
d
i
f
x
.
.
⋄
l
s
e
i
=
γ
∘
m
a
t
h
r
m
i
d
i
f
x
.
.
⋄
l
s
e
i
⊗
l
N
\gamma\circ r_{M}\otimes mathrm{id}{ifx..lse\par i}=\gamma\circ mathrm{id}{%
ifx..lse\par i}\otimes l_{N}
. Note that
ι
∘
(
α
⊗
β
)
𝜄
tensor-product
𝛼
𝛽
\iota\circ(\alpha\otimes\beta)
fits the right type, and we have
ι ∘ α ⊗ β ∘ r M ⊗ m a t h r m i d i f x . . ⋄ l s e i \displaystyle\iota\circ\alpha\otimes\beta\circ r_{M}\otimes mathrm{id}{ifx..%
lse\par i} = ι ∘ ( α ∘ r M ) ⊗ β absent tensor-product 𝜄 𝛼 subscript 𝑟 𝑀 𝛽 \displaystyle=\iota\circ(\alpha\circ r_{M})\otimes\beta
= ι ∘ ( r N ∘ α ⊗ g ) ⊗ β absent tensor-product 𝜄 tensor-product subscript 𝑟 𝑁 𝛼 𝑔 𝛽 \displaystyle=\iota\circ(r_{N}\circ\alpha\otimes g)\otimes\beta
= ι ∘ r N ⊗ m a t h r m i d i f x . . ⋄ l s e i ∘ ( α ⊗ g ) ⊗ β \displaystyle=\iota\circ r_{N}\otimes mathrm{id}{ifx..lse\par i}\circ(\alpha%
\otimes g)\otimes\beta
= ι ∘ m a t h r m i d i f x . . ⋄ l s e i ⊗ l Q ∘ ( α ⊗ g ) ⊗ β \displaystyle=\iota\circ mathrm{id}{ifx..lse\par i}\otimes l_{Q}\circ(\alpha%
\otimes g)\otimes\beta
= ι ∘ m a t h r m i d i f x . . ⋄ l s e i ⊗ l Q ∘ α ⊗ ( g ⊗ β ) \displaystyle=\iota\circ mathrm{id}{ifx..lse\par i}\otimes l_{Q}\circ\alpha%
\otimes(g\otimes\beta)
= ι ∘ α ⊗ ( l Q ∘ g ⊗ β ) absent tensor-product 𝜄 𝛼 tensor-product subscript 𝑙 𝑄 𝑔 𝛽 \displaystyle=\iota\circ\alpha\otimes(l_{Q}\circ g\otimes\beta)
= ι ∘ α ⊗ ( β ∘ l P ) absent tensor-product 𝜄 𝛼 𝛽 subscript 𝑙 𝑃 \displaystyle=\iota\circ\alpha\otimes(\beta\circ l_{P})
= ι ∘ α ⊗ β ∘ m a t h r m i d i f x . . ⋄ l s e i ⊗ l P \displaystyle=\iota\circ\alpha\otimes\beta\circ mathrm{id}{ifx..lse\par i}%
\otimes l_{P}
Moreover, the framing constraints on the square still hold up,
though this is a bit tedious to check.
When
𝒱
𝒱
\mathcal{V}
lacks Coequalisers , we
can still form a Virtual
Double Category of Bimodules , though the maps are very
tricky to define.
References