In category theory , a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback.
A traced symmetric monoidal category is a symmetric monoidal category C together with a family of functions
T
r
X
,
Y
U
:
C
(
X
⊗
U
,
Y
⊗
U
)
→
C
(
X
,
Y
)
{\displaystyle \mathrm {Tr} _{X,Y}^{U}:\mathbf {C} (X\otimes U,Y\otimes U)\to \mathbf {C} (X,Y)}
called a trace , satisfying the following conditions:
naturality in
X
{\displaystyle X}
: for every
f
:
X
⊗
U
→
Y
⊗
U
{\displaystyle f:X\otimes U\to Y\otimes U}
and
g
:
X
′
→
X
{\displaystyle g:X'\to X}
,
T
r
X
′
,
Y
U
(
f
∘
(
g
⊗
i
d
U
)
)
=
T
r
X
,
Y
U
(
f
)
∘
g
{\displaystyle \mathrm {Tr} _{X',Y}^{U}(f\circ (g\otimes \mathrm {id} _{U}))=\mathrm {Tr} _{X,Y}^{U}(f)\circ g}
naturality in
Y
{\displaystyle Y}
: for every
f
:
X
⊗
U
→
Y
⊗
U
{\displaystyle f:X\otimes U\to Y\otimes U}
and
g
:
Y
→
Y
′
{\displaystyle g:Y\to Y'}
,
T
r
X
,
Y
′
U
(
(
g
⊗
i
d
U
)
∘
f
)
=
g
∘
T
r
X
,
Y
U
(
f
)
{\displaystyle \mathrm {Tr} _{X,Y'}^{U}((g\otimes \mathrm {id} _{U})\circ f)=g\circ \mathrm {Tr} _{X,Y}^{U}(f)}
dinaturality in
U
{\displaystyle U}
: for every
f
:
X
⊗
U
→
Y
⊗
U
′
{\displaystyle f:X\otimes U\to Y\otimes U'}
and
g
:
U
′
→
U
{\displaystyle g:U'\to U}
T
r
X
,
Y
U
(
(
i
d
Y
⊗
g
)
∘
f
)
=
T
r
X
,
Y
U
′
(
f
∘
(
i
d
X
⊗
g
)
)
{\displaystyle \mathrm {Tr} _{X,Y}^{U}((\mathrm {id} _{Y}\otimes g)\circ f)=\mathrm {Tr} _{X,Y}^{U'}(f\circ (\mathrm {id} _{X}\otimes g))}
vanishing I: for every
f
:
X
⊗
I
→
Y
⊗
I
{\displaystyle f:X\otimes I\to Y\otimes I}
, (with
ρ
X
:
X
⊗
I
≅
X
{\displaystyle \rho _{X}\colon X\otimes I\cong X}
being the right unitor),
T
r
X
,
Y
I
(
f
)
=
ρ
Y
∘
f
∘
ρ
X
−
1
{\displaystyle \mathrm {Tr} _{X,Y}^{I}(f)=\rho _{Y}\circ f\circ \rho _{X}^{-1}}
vanishing II: for every
f
:
X
⊗
U
⊗
V
→
Y
⊗
U
⊗
V
{\displaystyle f:X\otimes U\otimes V\to Y\otimes U\otimes V}
T
r
X
,
Y
U
(
T
r
X
⊗
U
,
Y
⊗
U
V
(
f
)
)
=
T
r
X
,
Y
U
⊗
V
(
f
)
{\displaystyle \mathrm {Tr} _{X,Y}^{U}(\mathrm {Tr} _{X\otimes U,Y\otimes U}^{V}(f))=\mathrm {Tr} _{X,Y}^{U\otimes V}(f)}
superposing: for every
f
:
X
⊗
U
→
Y
⊗
U
{\displaystyle f:X\otimes U\to Y\otimes U}
and
g
:
W
→
Z
{\displaystyle g:W\to Z}
,
g
⊗
T
r
X
,
Y
U
(
f
)
=
T
r
W
⊗
X
,
Z
⊗
Y
U
(
g
⊗
f
)
{\displaystyle g\otimes \mathrm {Tr} _{X,Y}^{U}(f)=\mathrm {Tr} _{W\otimes X,Z\otimes Y}^{U}(g\otimes f)}
T
r
X
,
X
X
(
γ
X
,
X
)
=
i
d
X
{\displaystyle \mathrm {Tr} _{X,X}^{X}(\gamma _{X,X})=\mathrm {id} _{X}}
(where
γ
{\displaystyle \gamma }
is the symmetry of the monoidal category).
Properties [ ]
Every compact closed category admits a trace.
Given a traced monoidal category C , the Int construction generates the free (in some bicategorical sense) compact closure Int(C ) of C .
References [ ]
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