A Proposition Proposition (Matricial Representation) For f : ⊗ J X j − → ⊗ I Y i , there exists a unique family → Y i with f = � { f ij } i ∈ I , j ∈ J : X j − i ∈ I , j ∈ J ι i f ij ρ j , namely, f ij = ρ i f ι j . In particular, for | I | = m , | J | = n f 11 . . . f 1 n . . . . . . f = . . . f m 1 . . . f mn Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example 1 PInj, the category of sets and partial injective functions. ◮ X ⊗ Y = X ⊎ Y , Not a coproduct. Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example 1 PInj, the category of sets and partial injective functions. ◮ X ⊗ Y = X ⊎ Y , Not a coproduct. ◮ ρ j : ⊗ i ∈ I X i − → X j , ρ j ( x , i ) is undefined for i � = j and ρ j ( x , j ) = x , Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example 1 PInj, the category of sets and partial injective functions. ◮ X ⊗ Y = X ⊎ Y , Not a coproduct. ◮ ρ j : ⊗ i ∈ I X i − → X j , ρ j ( x , i ) is undefined for i � = j and ρ j ( x , j ) = x , ◮ ι j : X j − → ⊗ i ∈ I X i by ι j ( x ) = ( x , j ). Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example 2 Rel: The category of sets and binary relations. ◮ X ⊗ Y = X ⊎ Y , a biproduct, Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example 2 Rel: The category of sets and binary relations. ◮ X ⊗ Y = X ⊎ Y , a biproduct, ◮ ρ j : ⊗ i ∈ I X i − → X j , ρ j = { (( x , j ) , x ) | x ∈ X j } Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example 2 Rel: The category of sets and binary relations. ◮ X ⊗ Y = X ⊎ Y , a biproduct, ◮ ρ j : ⊗ i ∈ I X i − → X j , ρ j = { (( x , j ) , x ) | x ∈ X j } ◮ ι j : X j − → ⊗ i ∈ I X i , ι j = { ( x , ( x , j )) | x ∈ X j } = ρ op j . Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example 3: Hilb 2 ◮ Given a set X , Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example 3: Hilb 2 ◮ Given a set X , ◮ ℓ 2 ( X ): the set of all complex valued functions a on X for x ∈ X | a ( x ) | 2 is finite. which the (unordered) sum � Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example 3: Hilb 2 ◮ Given a set X , ◮ ℓ 2 ( X ): the set of all complex valued functions a on X for x ∈ X | a ( x ) | 2 is finite. which the (unordered) sum � ◮ ℓ 2 ( X ) is a Hilbert space ◮ || a || = ( � x ∈ X | a ( x ) | 2 ) 1 / 2 ◮ < a , b > = � x ∈ X a ( x ) b ( x ) for a , b ∈ ℓ 2 ( X ) Esfandiar Haghverdi On Categorical Models of GoILecture 1
◮ Barr’s ℓ 2 functor: contravariant faithful functor ℓ 2 : PInj op − → Hilb where Hilb is the category of Hilbert spaces and linear contractions (norm ≤ 1). Esfandiar Haghverdi On Categorical Models of GoILecture 1
◮ Barr’s ℓ 2 functor: contravariant faithful functor ℓ 2 : PInj op − → Hilb where Hilb is the category of Hilbert spaces and linear contractions (norm ≤ 1). 1. For a set X , ℓ 2 ( X ) is defined as above Esfandiar Haghverdi On Categorical Models of GoILecture 1
◮ Barr’s ℓ 2 functor: contravariant faithful functor ℓ 2 : PInj op − → Hilb where Hilb is the category of Hilbert spaces and linear contractions (norm ≤ 1). 1. For a set X , ℓ 2 ( X ) is defined as above 2. Given f : X − → Y in PInj, ℓ 2 ( f ) : ℓ 2 ( Y ) − → ℓ 2 ( X ) is defined by � b ( f ( x )) if x ∈ Dom ( f ), ℓ 2 ( f )( b )( x ) = 0 otherwise. Esfandiar Haghverdi On Categorical Models of GoILecture 1
◮ Barr’s ℓ 2 functor: contravariant faithful functor ℓ 2 : PInj op − → Hilb where Hilb is the category of Hilbert spaces and linear contractions (norm ≤ 1). 1. For a set X , ℓ 2 ( X ) is defined as above 2. Given f : X − → Y in PInj, ℓ 2 ( f ) : ℓ 2 ( Y ) − → ℓ 2 ( X ) is defined by � b ( f ( x )) if x ∈ Dom ( f ), ℓ 2 ( f )( b )( x ) = 0 otherwise. ◮ ℓ 2 ( X × Y ) ∼ = ℓ 2 ( X ) ⊗ ℓ 2 ( Y ) Esfandiar Haghverdi On Categorical Models of GoILecture 1
◮ Barr’s ℓ 2 functor: contravariant faithful functor ℓ 2 : PInj op − → Hilb where Hilb is the category of Hilbert spaces and linear contractions (norm ≤ 1). 1. For a set X , ℓ 2 ( X ) is defined as above 2. Given f : X − → Y in PInj, ℓ 2 ( f ) : ℓ 2 ( Y ) − → ℓ 2 ( X ) is defined by � b ( f ( x )) if x ∈ Dom ( f ), ℓ 2 ( f )( b )( x ) = 0 otherwise. ◮ ℓ 2 ( X × Y ) ∼ = ℓ 2 ( X ) ⊗ ℓ 2 ( Y ) ◮ ℓ 2 ( X ⊎ Y ) ∼ = ℓ 2 ( X ) ⊕ ℓ 2 ( Y ) Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example cont’d: Defining Hilb 2 ◮ Objects: ℓ 2 ( X ) for a set X Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example cont’d: Defining Hilb 2 ◮ Objects: ℓ 2 ( X ) for a set X ◮ Arrows: u : ℓ 2 ( X ) − → ℓ 2 ( Y ) is of the form ℓ 2 ( f ) for some partial injective function f : Y − → X Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example cont’d: Defining Hilb 2 ◮ Objects: ℓ 2 ( X ) for a set X ◮ Arrows: u : ℓ 2 ( X ) − → ℓ 2 ( Y ) is of the form ℓ 2 ( f ) for some partial injective function f : Y − → X ◮ For ℓ 2 ( X ) and ℓ 2 ( Y ) in Hilb 2 , the Hilbert space tensor product ℓ 2 ( X ) ⊗ ℓ 2 ( Y ) yields a tensor product in Hilb 2 . Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example cont’d: Defining Hilb 2 ◮ Objects: ℓ 2 ( X ) for a set X ◮ Arrows: u : ℓ 2 ( X ) − → ℓ 2 ( Y ) is of the form ℓ 2 ( f ) for some partial injective function f : Y − → X ◮ For ℓ 2 ( X ) and ℓ 2 ( Y ) in Hilb 2 , the Hilbert space tensor product ℓ 2 ( X ) ⊗ ℓ 2 ( Y ) yields a tensor product in Hilb 2 . ◮ Similarly for ℓ 2 ( X ) and ℓ 2 ( Y ) in Hilb 2 , the direct sum ℓ 2 ( X ) ⊕ ℓ 2 ( Y ) yields a tensor product ( not a coproduct) in Hilb 2 . Esfandiar Haghverdi On Categorical Models of GoILecture 1
The structure on PInj makes Hilb 2 into a UDC. ◮ { ℓ 2 ( f i ) } I ∈ Hilb 2 ( ℓ 2 ( X ) , ℓ 2 ( Y )), { f i } ∈ PInj( Y , X ), { ℓ 2 ( f i ) } is summable if { f i } is summable in PInj i ℓ 2 ( f i ) def ◮ � = ℓ 2 ( � i f i ) . Esfandiar Haghverdi On Categorical Models of GoILecture 1
Categorical trace (JSV 96) Definition A traced symmetric monoidal category is a symmetric monoidal category ( C , ⊗ , I , s ) with a family of functions Tr U X , Y : C ( X ⊗ U , Y ⊗ U ) − → C ( X , Y ) called a trace , subject to the following axioms: ◮ Natural in X , Tr U X , Y ( f ) g = Tr U X ′ , Y ( f ( g ⊗ 1 U )) where → Y ⊗ U , g : X ′ − f : X ⊗ U − → X , ◮ Natural in Y , gTr U X , Y ( f ) = Tr U X , Y ′ (( g ⊗ 1 U ) f ) where → Y ′ , f : X ⊗ U − → Y ⊗ U , g : Y − Esfandiar Haghverdi On Categorical Models of GoILecture 1
X , Y ((1 Y ⊗ g ) f ) = Tr U ′ ◮ Dinatural in U , Tr U X , Y ( f (1 X ⊗ g )) where → Y ⊗ U ′ , g : U ′ − f : X ⊗ U − → U , ◮ Vanishing (I,II) , Tr I X , Y ( f ) = f and Tr U ⊗ V X , Y ( g ) = Tr U X , Y ( Tr V X ⊗ U , Y ⊗ U ( g )) for f : X ⊗ I − → Y ⊗ I and g : X ⊗ U ⊗ V − → Y ⊗ U ⊗ V , ◮ Superposing , Tr U X , Y ( f ) ⊗ g = Tr U X ⊗ W , Y ⊗ Z ((1 Y ⊗ s U , Z )( f ⊗ g )(1 X ⊗ s W , U )) for f : X ⊗ U − → Y ⊗ U and g : W − → Z , ◮ Yanking , Tr U U , U ( s U , U ) = 1 U . Esfandiar Haghverdi On Categorical Models of GoILecture 1
Graphical Representation Y X X Y X’ X’ g g f f 1 U U U U U U Esfandiar Haghverdi On Categorical Models of GoILecture 1
1 1 X Y X Y X X Y Y f f g g U U U’ U U’ U Esfandiar Haghverdi On Categorical Models of GoILecture 1
X Y X Y f f U U V U V U V V Esfandiar Haghverdi On Categorical Models of GoILecture 1
U U U U U U Esfandiar Haghverdi On Categorical Models of GoILecture 1
W Z g W Z g X Y X Y f f U U U U Esfandiar Haghverdi On Categorical Models of GoILecture 1
Examples ◮ Consider the category FDVect k of finite dimensional vector spaces and linear transformations Esfandiar Haghverdi On Categorical Models of GoILecture 1
Examples ◮ Consider the category FDVect k of finite dimensional vector spaces and linear transformations ◮ Given f : V ⊗ U − → W ⊗ U , { v i } , { u j } , { w k } bases for V , U , W respectively. Esfandiar Haghverdi On Categorical Models of GoILecture 1
Examples ◮ Consider the category FDVect k of finite dimensional vector spaces and linear transformations ◮ Given f : V ⊗ U − → W ⊗ U , { v i } , { u j } , { w k } bases for V , U , W respectively. ◮ f ( v i ⊗ u j ) = � k , m a km w k ⊗ u m , ij Esfandiar Haghverdi On Categorical Models of GoILecture 1
Examples ◮ Consider the category FDVect k of finite dimensional vector spaces and linear transformations ◮ Given f : V ⊗ U − → W ⊗ U , { v i } , { u j } , { w k } bases for V , U , W respectively. ◮ f ( v i ⊗ u j ) = � k , m a km w k ⊗ u m , ij j , k a kj ◮ Tr U V , W ( f )( v i ) = � ij w k Esfandiar Haghverdi On Categorical Models of GoILecture 1
Examples ◮ Consider the category FDVect k of finite dimensional vector spaces and linear transformations ◮ Given f : V ⊗ U − → W ⊗ U , { v i } , { u j } , { w k } bases for V , U , W respectively. ◮ f ( v i ⊗ u j ) = � k , m a km w k ⊗ u m , ij j , k a kj ◮ Tr U V , W ( f )( v i ) = � ij w k ◮ This is just summing dim ( U ) many diagonal blocks, each of size dim ( W ) × dim ( V ) Esfandiar Haghverdi On Categorical Models of GoILecture 1
Examples ◮ Consider the category FDVect k of finite dimensional vector spaces and linear transformations ◮ Given f : V ⊗ U − → W ⊗ U , { v i } , { u j } , { w k } bases for V , U , W respectively. ◮ f ( v i ⊗ u j ) = � k , m a km w k ⊗ u m , ij j , k a kj ◮ Tr U V , W ( f )( v i ) = � ij w k ◮ This is just summing dim ( U ) many diagonal blocks, each of size dim ( W ) × dim ( V ) ◮ See what happens when dim ( V ) = dim ( W ) = 1, that is when V ∼ = W ∼ = k Esfandiar Haghverdi On Categorical Models of GoILecture 1
Examples, cont’d ◮ Consider the category Rel but with X ⊗ Y = X × Y Esfandiar Haghverdi On Categorical Models of GoILecture 1
Examples, cont’d ◮ Consider the category Rel but with X ⊗ Y = X × Y ◮ This is not a product, nor a coproduct. Esfandiar Haghverdi On Categorical Models of GoILecture 1
Examples, cont’d ◮ Consider the category Rel but with X ⊗ Y = X × Y ◮ This is not a product, nor a coproduct. ◮ Given R : X ⊗ U − → Y ⊗ U , Tr U X , Y ( R ) : X − → Y is defined by ( x , y ) ∈ Tr ( R ) iff ∃ u . ( x , u , y , u ) ∈ R . Esfandiar Haghverdi On Categorical Models of GoILecture 1
On Ubiquity of Trace ◮ Functional analysis and operator theory: Kadison & Ringrose Esfandiar Haghverdi On Categorical Models of GoILecture 1
On Ubiquity of Trace ◮ Functional analysis and operator theory: Kadison & Ringrose ◮ Knot Theory: Jones, Joyal, Street, Freyd, Yetter Esfandiar Haghverdi On Categorical Models of GoILecture 1
On Ubiquity of Trace ◮ Functional analysis and operator theory: Kadison & Ringrose ◮ Knot Theory: Jones, Joyal, Street, Freyd, Yetter ◮ Dimension theory of C ∗ -categories: Longo, Roberts Esfandiar Haghverdi On Categorical Models of GoILecture 1
On Ubiquity of Trace ◮ Functional analysis and operator theory: Kadison & Ringrose ◮ Knot Theory: Jones, Joyal, Street, Freyd, Yetter ◮ Dimension theory of C ∗ -categories: Longo, Roberts ◮ Action Calculi: Milner and Mifsud Esfandiar Haghverdi On Categorical Models of GoILecture 1
On Ubiquity of Trace ◮ Functional analysis and operator theory: Kadison & Ringrose ◮ Knot Theory: Jones, Joyal, Street, Freyd, Yetter ◮ Dimension theory of C ∗ -categories: Longo, Roberts ◮ Action Calculi: Milner and Mifsud ◮ Fixed Point and Iteration theory: Hasegawa, Haghverdi Esfandiar Haghverdi On Categorical Models of GoILecture 1
On Ubiquity of Trace ◮ Functional analysis and operator theory: Kadison & Ringrose ◮ Knot Theory: Jones, Joyal, Street, Freyd, Yetter ◮ Dimension theory of C ∗ -categories: Longo, Roberts ◮ Action Calculi: Milner and Mifsud ◮ Fixed Point and Iteration theory: Hasegawa, Haghverdi ◮ Cyclic Lambda Calculus: Hasegawa Esfandiar Haghverdi On Categorical Models of GoILecture 1
On Ubiquity of Trace ◮ Functional analysis and operator theory: Kadison & Ringrose ◮ Knot Theory: Jones, Joyal, Street, Freyd, Yetter ◮ Dimension theory of C ∗ -categories: Longo, Roberts ◮ Action Calculi: Milner and Mifsud ◮ Fixed Point and Iteration theory: Hasegawa, Haghverdi ◮ Cyclic Lambda Calculus: Hasegawa ◮ Asynchrony, Data flow networks: Selinger, Panangaden Esfandiar Haghverdi On Categorical Models of GoILecture 1
On Ubiquity of Trace ◮ Functional analysis and operator theory: Kadison & Ringrose ◮ Knot Theory: Jones, Joyal, Street, Freyd, Yetter ◮ Dimension theory of C ∗ -categories: Longo, Roberts ◮ Action Calculi: Milner and Mifsud ◮ Fixed Point and Iteration theory: Hasegawa, Haghverdi ◮ Cyclic Lambda Calculus: Hasegawa ◮ Asynchrony, Data flow networks: Selinger, Panangaden ◮ Geometry of Interaction: Abramsky, Haghverdi Esfandiar Haghverdi On Categorical Models of GoILecture 1
On Ubiquity of Trace ◮ Functional analysis and operator theory: Kadison & Ringrose ◮ Knot Theory: Jones, Joyal, Street, Freyd, Yetter ◮ Dimension theory of C ∗ -categories: Longo, Roberts ◮ Action Calculi: Milner and Mifsud ◮ Fixed Point and Iteration theory: Hasegawa, Haghverdi ◮ Cyclic Lambda Calculus: Hasegawa ◮ Asynchrony, Data flow networks: Selinger, Panangaden ◮ Geometry of Interaction: Abramsky, Haghverdi ◮ Models of MLL: Haghverdi Esfandiar Haghverdi On Categorical Models of GoILecture 1
Traced UDCs Proposition (Standard Trace Formula) Let C be a unique decomposition category such that for every → Y ⊗ U, the sum f 11 + � ∞ n =0 f 12 f n X , Y , U and f : X ⊗ U − 22 f 21 exists, where f ij are the components of f . Then, C is traced and ∞ � Tr U f 12 f n X , Y ( f ) = f 11 + 22 f 21 . n =0 ◮ Note that a UDC can be traced with a trace different from the standard one. Esfandiar Haghverdi On Categorical Models of GoILecture 1
Traced UDCs Proposition (Standard Trace Formula) Let C be a unique decomposition category such that for every → Y ⊗ U, the sum f 11 + � ∞ n =0 f 12 f n X , Y , U and f : X ⊗ U − 22 f 21 exists, where f ij are the components of f . Then, C is traced and ∞ � Tr U f 12 f n X , Y ( f ) = f 11 + 22 f 21 . n =0 ◮ Note that a UDC can be traced with a trace different from the standard one. ◮ In all my work, all traced UDCs are the ones with the standard trace. Esfandiar Haghverdi On Categorical Models of GoILecture 1
Examples: calculating traces Let C be a traced UDC. Then given any f : X ⊗ U − → Y ⊗ U , Tr U X , Y ( f ) exists. � g � 0 ◮ Let f : X ⊗ U − → Y ⊗ U be given by . Then h 0 �� g �� 0 Tr U X , Y ( f ) = Tr U n 00 n h = g + 0 h = = g + � X , Y h 0 g + 0 = g . � g � 0 ◮ Let f : X ⊗ U − → Y ⊗ U be given by . Then 0 h �� g �� 0 Tr U X , Y ( f ) = Tr U n 0 h n 0 = g + 0 = g . = g + � X , Y 0 h Esfandiar Haghverdi On Categorical Models of GoILecture 1
GoI Situation Definition A GoI Situation is a triple ( C , T , U ) where: ◮ C is a TSMC, Not necessarily a traced UDC! Esfandiar Haghverdi On Categorical Models of GoILecture 1
GoI Situation Definition A GoI Situation is a triple ( C , T , U ) where: ◮ C is a TSMC, Not necessarily a traced UDC! ◮ T : C − → C is a traced symmetric monoidal functor with the following retractions: 1. TT ✁ T ( e , e ′ ) (Comultiplication) 2. Id ✁ T ( d , d ′ ) (Dereliction) 3. T ⊗ T ✁ T ( c , c ′ ) (Contraction) 4. K I ✁ T ( w , w ′ ) (Weakening). Esfandiar Haghverdi On Categorical Models of GoILecture 1
GoI Situation Definition A GoI Situation is a triple ( C , T , U ) where: ◮ C is a TSMC, Not necessarily a traced UDC! ◮ T : C − → C is a traced symmetric monoidal functor with the following retractions: 1. TT ✁ T ( e , e ′ ) (Comultiplication) 2. Id ✁ T ( d , d ′ ) (Dereliction) 3. T ⊗ T ✁ T ( c , c ′ ) (Contraction) 4. K I ✁ T ( w , w ′ ) (Weakening). ◮ U a reflexive object of C : 1. U ⊗ U ✁ U ( j , k ) 2. I ✁ U 3. TU ✁ U ( u , v ) Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example: PInj ◮ In PInj we let ⊗ = ⊎ , ◮ The tensor unit is the empty set ∅ . ◮ T = N × − , with T = ( T , ψ, ψ I ): ψ X , Y : N × X ⊎ N × Y − → N × ( X ⊎ Y ) given by (1 , ( n , x )) �→ ( n , (1 , x )) and (2 , ( n , y )) �→ ( n , (2 , y )). ψ has an inverse defined by: ( n , (1 , x )) �→ (1 , ( n , x )) and ( n , (2 , y )) �→ (2 , ( n , y )). ψ I : ∅ − → N × ∅ given by 1 ∅ . Esfandiar Haghverdi On Categorical Models of GoILecture 1
◮ T is additive, and thus it is also traced: Given f : X ⊎ U − → Y ⊎ U : X , Y ( f ) = Tr N × U N × X , N × Y ( ψ − 1 (1 N × f ) ψ ). 1 N × Tr U ◮ N is a reflexive object. 1. N ⊎ N ✁ N ( j , k ) is given as follows: j : N ⊎ N − → N , j (1 , n ) = 2 n , j (2 , n ) = 2 n + 1 and k : N − → N ⊎ N , k ( n ) = (1 , n / 2) for n even, and (2 , ( n − 1) / 2) for n odd. Esfandiar Haghverdi On Categorical Models of GoILecture 1
◮ T is additive, and thus it is also traced: Given f : X ⊎ U − → Y ⊎ U : X , Y ( f ) = Tr N × U N × X , N × Y ( ψ − 1 (1 N × f ) ψ ). 1 N × Tr U ◮ N is a reflexive object. 1. N ⊎ N ✁ N ( j , k ) is given as follows: j : N ⊎ N − → N , j (1 , n ) = 2 n , j (2 , n ) = 2 n + 1 and k : N − → N ⊎ N , k ( n ) = (1 , n / 2) for n even, and (2 , ( n − 1) / 2) for n odd. 2. ∅ ✁ N using the empty partial function as the retract morphisms. Esfandiar Haghverdi On Categorical Models of GoILecture 1
◮ T is additive, and thus it is also traced: Given f : X ⊎ U − → Y ⊎ U : X , Y ( f ) = Tr N × U N × X , N × Y ( ψ − 1 (1 N × f ) ψ ). 1 N × Tr U ◮ N is a reflexive object. 1. N ⊎ N ✁ N ( j , k ) is given as follows: j : N ⊎ N − → N , j (1 , n ) = 2 n , j (2 , n ) = 2 n + 1 and k : N − → N ⊎ N , k ( n ) = (1 , n / 2) for n even, and (2 , ( n − 1) / 2) for n odd. 2. ∅ ✁ N using the empty partial function as the retract morphisms. 3. N × N ✁ N ( u , v ) is defined as: u ( m , n ) = < m , n > = ( m + n +1)( m + n ) + n (Cantor surjective 2 pairing) and v as its inverse, v ( n ) = ( n 1 , n 2 ) with < n 1 , n 2 > = n . Esfandiar Haghverdi On Categorical Models of GoILecture 1
PInj cont’d We next define the necessary monoidal natural transformations. e ′ e X X ◮ N × ( N × X ) − → N × X and N × X − → N × ( N × X ) Esfandiar Haghverdi On Categorical Models of GoILecture 1
PInj cont’d We next define the necessary monoidal natural transformations. e ′ e X X ◮ N × ( N × X ) − → N × X and N × X − → N × ( N × X ) e X ◮ N × ( N × X ) − → N × X is defined by, e X ( n 1 , ( n 2 , x )) = ( < n 1 , n 2 >, x ). Esfandiar Haghverdi On Categorical Models of GoILecture 1
PInj cont’d We next define the necessary monoidal natural transformations. e ′ e X X ◮ N × ( N × X ) − → N × X and N × X − → N × ( N × X ) e X ◮ N × ( N × X ) − → N × X is defined by, e X ( n 1 , ( n 2 , x )) = ( < n 1 , n 2 >, x ). d ′ d X ◮ X X − → N × X and N × X − → X d X ( x ) = ( n 0 , x ) for a fixed n 0 ∈ N . Esfandiar Haghverdi On Categorical Models of GoILecture 1
PInj cont’d We next define the necessary monoidal natural transformations. e ′ e X X ◮ N × ( N × X ) − → N × X and N × X − → N × ( N × X ) e X ◮ N × ( N × X ) − → N × X is defined by, e X ( n 1 , ( n 2 , x )) = ( < n 1 , n 2 >, x ). d ′ d X ◮ X X − → N × X and N × X − → X d X ( x ) = ( n 0 , x ) for a fixed n 0 ∈ N . ◮ � x , if n = n 0 ; d ′ X ( n , x ) = undefined, else. Esfandiar Haghverdi On Categorical Models of GoILecture 1
c X ◮ ( N × X ) ⊎ ( N × X ) − → N × X and c ′ X N × X − → ( N × X ) ⊎ ( N × X ). � (1 , ( n , x )) �→ (2 n , x ) c X = (2 , ( n , x )) �→ (2 n + 1 , x ) � (1 , ( n / 2 , x )) , if n is even; c ′ X ( n , x ) = (2 , (( n − 1) / 2 , x )) , if n is odd. Esfandiar Haghverdi On Categorical Models of GoILecture 1
c X ◮ ( N × X ) ⊎ ( N × X ) − → N × X and c ′ X N × X − → ( N × X ) ⊎ ( N × X ). � (1 , ( n , x )) �→ (2 n , x ) c X = (2 , ( n , x )) �→ (2 n + 1 , x ) � (1 , ( n / 2 , x )) , if n is even; c ′ X ( n , x ) = (2 , (( n − 1) / 2 , x )) , if n is odd. w ′ w X ◮ ∅ X − → N × X and N × X − → ∅ . Esfandiar Haghverdi On Categorical Models of GoILecture 1
Example: Traced UDC based ◮ ( PInj , N × − , N ) ◮ ( Hilb 2 , ℓ 2 ⊗ − , ℓ 2 ) ◮ ( Rel ⊕ , N × − , N ) ◮ ( Pfn , N × − , N ) Esfandiar Haghverdi On Categorical Models of GoILecture 1
GoI Interpretation Recall that in categorical denotational semantics: ◮ We are given a logical system L to model, e.g. IL ◮ We are given a model category C with enough structure, e.g. a CCC, ◮ Formulas are interpreted as objects ◮ Proofs are intepreted as morphisms, indeed morphisms are equivalence classes of proofs ◮ Cut-elimination (proof transformation) is interpreted by provable equality. ◮ One proves a soundness theorem: Theorem Given a sequent Γ ⊢ A and proofs Π and Π ′ such that Π ≻ Π ′ , Π ′ then Π = : Γ − → A . Esfandiar Haghverdi On Categorical Models of GoILecture 1
GoI interpretation In GoI interpretation: ◮ We are given a logical system L to model, e.g. MLL, ◮ We are given a GoI Situation ( C , T , U ), e.g. ( PInj , N × − , N ), ◮ Formulas are interpreted as types (see below), ◮ Proofs are interpreted as morphisms in C ( U , U ), ◮ Cut-elimination (proof transformation) is interpreted by the execution formula Esfandiar Haghverdi On Categorical Models of GoILecture 1
◮ One proves a finiteness theorem Theorem Given a sequent Γ ⊢ A with a proof Π and cut formulas represented by σ , then EX ( θ (Π) , σ ) exists. Esfandiar Haghverdi On Categorical Models of GoILecture 1
◮ One proves a finiteness theorem Theorem Given a sequent Γ ⊢ A with a proof Π and cut formulas represented by σ , then EX ( θ (Π) , σ ) exists. ◮ And a soundness theorem Theorem Given a sequent Γ ⊢ A and proofs Π and Π ′ such that Π ≻ Π ′ , then EX ( θ (Π) , σ ) = EX ( θ (Π ′ ) , τ ) where σ and τ represent the cut formulas in Π and Π ′ respectively (see below). Esfandiar Haghverdi On Categorical Models of GoILecture 1
GoI Interpretation: proofs Hereafter we shall be working with traced UDCs. ◮ Π a proof of ⊢ [∆] , Γ, | ∆ | = 2 m and | Γ | = n . ◮ ∆ keeps track of the cut formulas, e.g., ∆ = A , A ⊥ , B , B ⊥ , ◮ θ (Π) : U n +2 m − → U n +2 m ◮ σ : U 2 m − → U 2 m = s ⊗ m U , U Γ Γ θ(Π) ∆ ∆ Esfandiar Haghverdi On Categorical Models of GoILecture 1
GoI Int, cont’d axiom : ⊢ A , A ⊥ , m = 0 , n = 2. θ (Π) = s U , U . A A A A Esfandiar Haghverdi On Categorical Models of GoILecture 1
cut : Π ′ Π ′′ . . . . . . ⊢ [∆ ′ ] , Γ ′ , A ⊢ [∆ ′′ ] , A ⊥ , Γ ′′ ( cut ) ⊢ [∆ ′ , ∆ ′′ , A , A ⊥ ] , Γ ′ , Γ ′′ τ −1 τ θ(Π ’ ) θ(Π ’’ ) Esfandiar Haghverdi On Categorical Models of GoILecture 1
times : Recall U ⊗ U ✁ U ( j , k ) Π ′ Π ′′ . . . . . . ⊢ [∆ ′ ] , Γ ′ , A ⊢ [∆ ′′ ] , Γ ′′ , B ( times ) ⊢ [∆ ′ , ∆ ′′ ] , Γ ′ , Γ ′′ , A ⊗ B τ −1 τ f g θ(Π ’ ) j1 k1 j2 k2 θ(Π ’’ ) Esfandiar Haghverdi On Categorical Models of GoILecture 1
of course : Recall TU ✁ U ( u , v ) and TT ✁ T ( e , e ′ ) Π ′ . . . ⊢ [∆] , ?Γ ′ , A ⊢ [∆] , ?Γ ′ , ! A ( ofcourse ) T eU v e’ v u u U θ(Π ’) v u v u Esfandiar Haghverdi On Categorical Models of GoILecture 1
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