Representation of Partial Traces
TACL 2015, Ischia, Italy
Marc Bagnol — University of Ottawa
Traces in symmetric monoidal categories
Monoidal: a category with an associative bifunctor ⊗ and a unit object 1 .
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Traces in symmetric monoidal categories
Monoidal: a category with an associative bifunctor ⊗ and a unit object 1 .
Symmetric: moreover has natural isomorphisms σA,B : A ⊗ B → B ⊗ A ,
such that σA,B ◦ σB,A = Id A⊗ B .
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Traces in symmetric monoidal categories
Monoidal: a category with an associative bifunctor ⊗ and a unit object 1 .
Symmetric: moreover has natural isomorphisms σA,B : A ⊗ B → B ⊗ A ,
such that σA,B ◦ σB,A = Id A⊗ B .
Trace (A. Joyal, R. Street, D. Verity): operation turning f : A ⊗ U → B ⊗ U
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Traces in symmetric monoidal categories
Monoidal: a category with an associative bifunctor ⊗ and a unit object 1 .
Symmetric: moreover has natural isomorphisms σA,B : A ⊗ B → B ⊗ A ,
such that σA,B ◦ σB,A = Id A⊗ B .
Trace (A. Joyal, R. Street, D. Verity): operation turning f : A ⊗ U → B ⊗ U
into TrU [ f ] : A → B .
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Traces in symmetric monoidal categories
Monoidal: a category with an associative bifunctor ⊗ and a unit object 1 .
Symmetric: moreover has natural isomorphisms σA,B : A ⊗ B → B ⊗ A ,
such that σA,B ◦ σB,A = Id A⊗ B .
Trace (A. Joyal, R. Street, D. Verity): operation turning f : A ⊗ U → B ⊗ U
into TrU [ f ] : A → B .
U
U
f
A
B
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Traces in symmetric monoidal categories
Monoidal: a category with an associative bifunctor ⊗ and a unit object 1 .
Symmetric: moreover has natural isomorphisms σA,B : A ⊗ B → B ⊗ A ,
such that σA,B ◦ σB,A = Id A⊗ B .
Trace (A. Joyal, R. Street, D. Verity): operation turning f : A ⊗ U → B ⊗ U
into TrU [ f ] : A → B .
U
U
f
A
B
Understood as a feedback along U .
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Traces in symmetric monoidal categories
Monoidal: a category with an associative bifunctor ⊗ and a unit object 1 .
Symmetric: moreover has natural isomorphisms σA,B : A ⊗ B → B ⊗ A ,
such that σA,B ◦ σB,A = Id A⊗ B .
Trace (A. Joyal, R. Street, D. Verity): operation turning f : A ⊗ U → B ⊗ U
into TrU [ f ] : A → B .
U
U
f
A
B
Understood as a feedback along U .
Ubiquitous structure in mathematics: linear algebra, topology, knot theory,
proof theory. . .
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Partial traces
P. Scott & E. Haghverdi: axiomatization of partially-defined trace,
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Partial traces
P. Scott & E. Haghverdi: axiomatization of partially-defined trace, capturing
the idea of (partially defined) categorical feedback.
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Partial traces
P. Scott & E. Haghverdi: axiomatization of partially-defined trace, capturing
the idea of (partially defined) categorical feedback.
One example of partial traces axiom: sliding
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Partial traces
P. Scott & E. Haghverdi: axiomatization of partially-defined trace, capturing
the idea of (partially defined) categorical feedback.
One example of partial traces axiom: sliding
0
TrU f (Id A ⊗ g) TrU (IdB ⊗ g) f
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Partial traces
P. Scott & E. Haghverdi: axiomatization of partially-defined trace, capturing
the idea of (partially defined) categorical feedback.
One example of partial traces axiom: sliding
0
TrU f (Id A ⊗ g) TrU (IdB ⊗ g) f
U
A
a
g
U0
f
U0
U
B
A
f
U
g
U0
B
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Partial traces and sub-categories
A straightforward way to build partial traces:
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Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
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Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
◦ Take any sub-symmetric monoidal category C ⊆ D .
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Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
◦ Take any sub-symmetric monoidal category C ⊆ D .
◦ If f : A ⊗ U → B ⊗ U is in C ,
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Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
◦ Take any sub-symmetric monoidal category C ⊆ D .
◦ If f : A ⊗ U → B ⊗ U is in C , it always has a trace TrU [ f ] in D .
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Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
◦ Take any sub-symmetric monoidal category C ⊆ D .
◦ If f : A ⊗ U → B ⊗ U is in C , it always has a trace TrU [ f ] in D .
( TrU [ f ] may or may not end up in C )
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Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
◦ Take any sub-symmetric monoidal category C ⊆ D .
◦ If f : A ⊗ U → B ⊗ U is in C , it always has a trace TrU [ f ] in D .
( TrU [ f ] may or may not end up in C )
b on C as:
Define a partial trace Tr
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Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
◦ Take any sub-symmetric monoidal category C ⊆ D .
◦ If f : A ⊗ U → B ⊗ U is in C , it always has a trace TrU [ f ] in D .
( TrU [ f ] may or may not end up in C )
b on C as:
Define a partial trace Tr
if TrU [ f ] ∈ C
4/9
Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
◦ Take any sub-symmetric monoidal category C ⊆ D .
◦ If f : A ⊗ U → B ⊗ U is in C , it always has a trace TrU [ f ] in D .
( TrU [ f ] may or may not end up in C )
b on C as:
Define a partial trace Tr
b U [ f ] = TrU [ f ]
if TrU [ f ] ∈ C then Tr
4/9
Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
◦ Take any sub-symmetric monoidal category C ⊆ D .
◦ If f : A ⊗ U → B ⊗ U is in C , it always has a trace TrU [ f ] in D .
( TrU [ f ] may or may not end up in C )
b on C as:
Define a partial trace Tr
b U [ f ] = TrU [ f ] , undefined otherwise
if TrU [ f ] ∈ C then Tr
4/9
Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
◦ Take any sub-symmetric monoidal category C ⊆ D .
◦ If f : A ⊗ U → B ⊗ U is in C , it always has a trace TrU [ f ] in D .
( TrU [ f ] may or may not end up in C )
b on C as:
Define a partial trace Tr
b U [ f ] = TrU [ f ] , undefined otherwise
if TrU [ f ] ∈ C then Tr
Does any partial trace arise this way?
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Partial traces and sub-categories
A straightforward way to build partial traces:
◦ Consider a totally traced category D .
◦ Take any sub-symmetric monoidal category C ⊆ D .
◦ If f : A ⊗ U → B ⊗ U is in C , it always has a trace TrU [ f ] in D .
( TrU [ f ] may or may not end up in C )
b on C as:
Define a partial trace Tr
b U [ f ] = TrU [ f ] , undefined otherwise
if TrU [ f ] ∈ C then Tr
Does any partial trace arise this way?
O. Malherbe, P. Scott, P. Selinger: representation theorem.
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The representation theorem
More precisely: any partially traced category embeds in a totally traced one.
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The representation theorem
More precisely: any partially traced category embeds in a totally traced one.
We also have a universal property:
C
EC
T(C)
(where C is partially traced, T(C) is the totally traced category in which it embeds,
D is any other totally traced category, with F a traced functor from C to D )
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The representation theorem
More precisely: any partially traced category embeds in a totally traced one.
We also have a universal property:
C
F
D
(where C is partially traced, T(C) is the totally traced category in which it embeds,
D is any other totally traced category, with F a traced functor from C to D )
5/9
The representation theorem
More precisely: any partially traced category embeds in a totally traced one.
We also have a universal property:
C
EC
T(C)
F
D
(where C is partially traced, T(C) is the totally traced category in which it embeds,
D is any other totally traced category, with F a traced functor from C to D )
5/9
The representation theorem
More precisely: any partially traced category embeds in a totally traced one.
We also have a universal property:
C
EC
T(C)
G
F
D
(where C is partially traced, T(C) is the totally traced category in which it embeds,
D is any other totally traced category, with F a traced functor from C to D )
5/9
The representation theorem
More precisely: any partially traced category embeds in a totally traced one.
We also have a universal property:
C
EC
T(C)
G
F
D
(where C is partially traced, T(C) is the totally traced category in which it embeds,
D is any other totally traced category, with F a traced functor from C to D )
Original proof: intermediate partial version of the Int(·) construction and
“paracategories”.
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The representation theorem
More precisely: any partially traced category embeds in a totally traced one.
We also have a universal property:
C
EC
T(C)
G
F
D
(where C is partially traced, T(C) is the totally traced category in which it embeds,
D is any other totally traced category, with F a traced functor from C to D )
Original proof: intermediate partial version of the Int(·) construction and
“paracategories”.
Contribution: a more direct and simplified proof.
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The proof (I): the dialect construction
A generic construction D(C) on any monoidal category C .
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The proof (I): the dialect construction
A generic construction D(C) on any monoidal category C .
Basic idea: add a “state space” to morphisms.
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The proof (I): the dialect construction
A generic construction D(C) on any monoidal category C .
Basic idea: add a “state space” to morphisms.
A morphism from A to B in D(C) is a pair ( f , U ) with
6/9
The proof (I): the dialect construction
A generic construction D(C) on any monoidal category C .
Basic idea: add a “state space” to morphisms.
A morphism from A to B in D(C) is a pair ( f , U ) with
◦ U an object of C .
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The proof (I): the dialect construction
A generic construction D(C) on any monoidal category C .
Basic idea: add a “state space” to morphisms.
A morphism from A to B in D(C) is a pair ( f , U ) with
◦ U an object of C .
◦ f : A ⊗ U → B ⊗ U a morphism of C .
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The proof (I): the dialect construction
A generic construction D(C) on any monoidal category C .
Basic idea: add a “state space” to morphisms.
A morphism from A to B in D(C) is a pair ( f , U ) with
◦ U an object of C .
◦ f : A ⊗ U → B ⊗ U a morphism of C .
When composing ( f , U ) and ( g, V ) the state spaces do not interact.
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The proof (II): hiding and congruences
Hiding: given a partially traced C we can look at D(C) and define a hiding
operation
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The proof (II): hiding and congruences
Hiding: given a partially traced C we can look at D(C) and define a hiding
operation turning ( f , V ) : A ⊗ U → B ⊗ U into
7/9
The proof (II): hiding and congruences
Hiding: given a partially traced C we can look at D(C) and define a hiding
operation turning ( f , V ) : A ⊗ U → B ⊗ U into
HU [ f , V ] = ( f , U ⊗ V ) : A → B
7/9
The proof (II): hiding and congruences
Hiding: given a partially traced C we can look at D(C) and define a hiding
operation turning ( f , V ) : A ⊗ U → B ⊗ U into
HU [ f , V ] = ( f , U ⊗ V ) : A → B
H [·] behaves a lot like a (total) trace.
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The proof (II): hiding and congruences
Hiding: given a partially traced C we can look at D(C) and define a hiding
operation turning ( f , V ) : A ⊗ U → B ⊗ U into
HU [ f , V ] = ( f , U ⊗ V ) : A → B
H [·] behaves a lot like a (total) trace.
Congruences: consider the equivalence relation on morphisms generated by
some required equations, including
7/9
The proof (II): hiding and congruences
Hiding: given a partially traced C we can look at D(C) and define a hiding
operation turning ( f , V ) : A ⊗ U → B ⊗ U into
HU [ f , V ] = ( f , U ⊗ V ) : A → B
H [·] behaves a lot like a (total) trace.
Congruences: consider the equivalence relation on morphisms generated by
some required equations, including
( f , U ⊗ V ) ≈ (TrV [ f ], U ) when TrV [ f ] is defined.
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The proof (II): hiding and congruences
Hiding: given a partially traced C we can look at D(C) and define a hiding
operation turning ( f , V ) : A ⊗ U → B ⊗ U into
HU [ f , V ] = ( f , U ⊗ V ) : A → B
H [·] behaves a lot like a (total) trace.
Congruences: consider the equivalence relation on morphisms generated by
some required equations, including
( f , U ⊗ V ) ≈ (TrV [ f ], U ) when TrV [ f ] is defined.
Then we can set T(C) = D(C) / ≈
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The proof (II): hiding and congruences
Hiding: given a partially traced C we can look at D(C) and define a hiding
operation turning ( f , V ) : A ⊗ U → B ⊗ U into
HU [ f , V ] = ( f , U ⊗ V ) : A → B
H [·] behaves a lot like a (total) trace.
Congruences: consider the equivalence relation on morphisms generated by
some required equations, including
( f , U ⊗ V ) ≈ (TrV [ f ], U ) when TrV [ f ] is defined.
Then we can set T(C) = D(C) / ≈ in which H [·] induces a total trace,
encompassing the original partial trace of C .
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The proof (III): a sketch
We can embed C in T(C) by setting EC ( f ) = ( f , 1) .
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The proof (III): a sketch
We can embed C in T(C) by setting EC ( f ) = ( f , 1) .
Is it really an embedding?
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The proof (III): a sketch
We can embed C in T(C) by setting EC ( f ) = ( f , 1) .
Is it really an embedding? We check that ( f , 1) ≈ ( g, 1) implies f = g .
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The proof (III): a sketch
We can embed C in T(C) by setting EC ( f ) = ( f , 1) .
Is it really an embedding? We check that ( f , 1) ≈ ( g, 1) implies f = g .
Because ≈ is freely generated, we can do it by induction on chains of
elementary equivalences.
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The proof (III): a sketch
We can embed C in T(C) by setting EC ( f ) = ( f , 1) .
Is it really an embedding? We check that ( f , 1) ≈ ( g, 1) implies f = g .
Because ≈ is freely generated, we can do it by induction on chains of
elementary equivalences.
Universal property:
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The proof (III): a sketch
We can embed C in T(C) by setting EC ( f ) = ( f , 1) .
Is it really an embedding? We check that ( f , 1) ≈ ( g, 1) implies f = g .
Because ≈ is freely generated, we can do it by induction on chains of
elementary equivalences.
C
EC
T(C)
Universal property: we can close the diagram
F
D
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The proof (III): a sketch
We can embed C in T(C) by setting EC ( f ) = ( f , 1) .
Is it really an embedding? We check that ( f , 1) ≈ ( g, 1) implies f = g .
Because ≈ is freely generated, we can do it by induction on chains of
elementary equivalences.
C
EC
T(C)
Universal property: we can close the diagram
G
F
D
by setting G ( f , U ) = Tr
FU
[F f ] .
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The proof (III): a sketch
We can embed C in T(C) by setting EC ( f ) = ( f , 1) .
Is it really an embedding? We check that ( f , 1) ≈ ( g, 1) implies f = g .
Because ≈ is freely generated, we can do it by induction on chains of
elementary equivalences.
C
EC
T(C)
Universal property: we can close the diagram
G
F
D
by setting G ( f , U ) = Tr
FU
[F f ] .
(well defined because ( f , U ) ≈ ( g, V ) implies TrFU (F f ) = TrFV (Fg) )
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Conclusion
Easier proof of an already known result: the representation theorem for
partially traced categories.
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Conclusion
Easier proof of an already known result: the representation theorem for
partially traced categories.
Allows intuitive diagrammatic reasoning also in the partially-defined case.
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Conclusion
Easier proof of an already known result: the representation theorem for
partially traced categories.
Allows intuitive diagrammatic reasoning also in the partially-defined case.
A question: how do the constructions of both proofs relate?
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Conclusion
Easier proof of an already known result: the representation theorem for
partially traced categories.
Allows intuitive diagrammatic reasoning also in the partially-defined case.
A question: how do the constructions of both proofs relate?
Another: can the setting be tweaked to account for partial traces of
infinite-dimensional Hilbert spaces and C∗-algebras?
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Conclusion
Easier proof of an already known result: the representation theorem for
partially traced categories.
Allows intuitive diagrammatic reasoning also in the partially-defined case.
A question: how do the constructions of both proofs relate?
Another: can the setting be tweaked to account for partial traces of
infinite-dimensional Hilbert spaces and C∗-algebras?
. . . Thank you for your attention
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