Compact closed categories An easy way to get higher-order processes is to use compact closed categories: Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 15 / 47
Compact closed categories An easy way to get higher-order processes is to use compact closed categories: Definition An SMC C is compact closed if every object A has a dual object A ∗ , i.e. there exists η A : I → A ∗ ⊗ A and ǫ A : A ⊗ A ∗ → I , satisfying: ( ǫ A ⊗ 1 A ) ◦ (1 A ⊗ η A ) = 1 A (1 A ∗ ⊗ ǫ A ) ◦ ( η A ⊗ 1 A ∗ ) = 1 A ∗ A ∗ A = = A ∗ A A A ∗ A A ∗ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 15 / 47
Higher-order processes Processes send states to states: �→ f ρ ρ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 16 / 47
Higher-order processes Processes send states to states: �→ f ρ ρ In compact closed categories, everything is a state, thanks to process-state duality : : A ∗ ⊗ B : A ⊸ B ↔ f f ρ f Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 16 / 47
Higher-order processes Processes send states to states: �→ f ρ ρ In compact closed categories, everything is a state, thanks to process-state duality : : A ∗ ⊗ B : A ⊸ B ↔ f f ρ f ⇒ higher order processes are the same as first-order processes : w : ( A ⊸ B ) ⊸ ( C ⊸ D ) �→ f f Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 16 / 47
Some handy notation We can treat everything as a state, and write states in any shape we like: D C ∗ D A ∗ C B := w w B A Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 17 / 47
Some handy notation We can treat everything as a state, and write states in any shape we like: D C ∗ D A ∗ C B := w w B A Then plugging shapes together means composing the appropriate caps: D A ∗ D C B C ∗ B ∗ C := w Φ w Φ B A Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 17 / 47
Some handy notation It looks like we can now freely work with higher-order causal processes: D C w B : A ⊸ ( B ⊸ C ) ⊸ D Y v X A ...but theres a problem. Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 18 / 47
The compact collapse In a compact closed category: ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 19 / 47
The compact collapse In a compact closed category: ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ Which gives: ( A ⊸ B ) ⊸ C Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 19 / 47
The compact collapse In a compact closed category: ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ Which gives: ( A ⊸ B ) ∗ ⊗ C ∼ ( A ⊸ B ) ⊸ C = Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 19 / 47
The compact collapse In a compact closed category: ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ Which gives: ( A ⊸ B ) ∗ ⊗ C ∼ ( A ⊸ B ) ⊸ C = ( A ∗ ⊗ B ) ∗ ⊗ C ∼ = Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 19 / 47
The compact collapse In a compact closed category: ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ Which gives: ( A ⊸ B ) ∗ ⊗ C ∼ ( A ⊸ B ) ⊸ C = ( A ∗ ⊗ B ) ∗ ⊗ C ∼ = A ⊗ B ∗ ⊗ C ∼ = Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 19 / 47
The compact collapse In a compact closed category: ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ Which gives: ( A ⊸ B ) ∗ ⊗ C ∼ ( A ⊸ B ) ⊸ C = ( A ∗ ⊗ B ) ∗ ⊗ C ∼ = A ⊗ B ∗ ⊗ C ∼ = B ∗ ⊗ A ⊗ C ∼ = Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 19 / 47
The compact collapse In a compact closed category: ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ Which gives: ( A ⊸ B ) ∗ ⊗ C ∼ ( A ⊸ B ) ⊸ C = ( A ∗ ⊗ B ) ∗ ⊗ C ∼ = A ⊗ B ∗ ⊗ C ∼ = B ∗ ⊗ A ⊗ C ∼ = ∼ = B ⊸ A ⊗ C Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 19 / 47
The compact collapse In a compact closed category: ( A ⊗ B ) ∗ = A ∗ ⊗ B ∗ Which gives: ( A ⊸ B ) ∗ ⊗ C ∼ ( A ⊸ B ) ⊸ C = ( A ∗ ⊗ B ) ∗ ⊗ C ∼ = A ⊗ B ∗ ⊗ C ∼ = B ∗ ⊗ A ⊗ C ∼ = ∼ = B ⊸ A ⊗ C ⇒ everything collapses to first order! Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 19 / 47
The compact collapse But first-order causal � = second-order causal: ∀ Φ causal . = w Φ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 20 / 47
The compact collapse But first-order causal � = second-order causal: ∀ Φ causal . = w Φ So, causal types are richer than compact-closed types. In particular: A ⊸ B := ( A ⊗ B ∗ ) ∗ �∼ = A ∗ ⊗ B Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 20 / 47
The compact collapse But first-order causal � = second-order causal: ∀ Φ causal . = w Φ So, causal types are richer than compact-closed types. In particular: A ⊸ B := ( A ⊗ B ∗ ) ∗ �∼ = A ∗ ⊗ B If we drop this iso from the definition of compact closed, we get a ∗ -autonomous category . Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 20 / 47
Definition A ∗ -autonomous category is a symmetric monoidal category equipped with a full and faithful functor ( − ) ∗ : C op → C such that, by letting: A ⊸ B := ( A ⊗ B ∗ ) ∗ (1) there exists a natural isomorphism: C ( A ⊗ B , C ) ∼ = C ( A , B ⊸ C ) (2) Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 21 / 47
The recipe Precausal category C Caus [ C ] �→ ∗ -autonomous category compact closed category of ‘raw materials’ capturing ‘logic of causality’ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 22 / 47
The recipe Precausal category C Caus [ C ] �→ ∗ -autonomous category compact closed category of ‘raw materials’ capturing ‘logic of causality’ Mat ( R + ) �→ higher-order stochastic maps �→ higher-order quantum channels CPM Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 22 / 47
Precausal categories Precausal categories give ‘good’ raw materials, i.e. discarding behaves well w.r.t. the categorical structure. The standard examples are Mat ( R + ) and CPM . Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 23 / 47
Precausal categories Precausal categories give ‘good’ raw materials, i.e. discarding behaves well w.r.t. the categorical structure. The standard examples are Mat ( R + ) and CPM . Definition A precausal category is a compact closed category C such that: ( C1 ) C has discarding processes for every system ( C2 ) For every (non-zero) system A , the dimension of A : d A := A is an invertible scalar. Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 23 / 47
Precausal categories Precausal categories give ‘good’ raw materials, i.e. discarding behaves well w.r.t. the categorical structure. The standard examples are Mat ( R + ) and CPM . Definition A precausal category is a compact closed category C such that: ( C1 ) C has discarding processes for every system ( C2 ) For every (non-zero) system A , the dimension of A : d A := A is an invertible scalar. ( C3 ) C has enough causal states ( C4 ) Second-order causal processes factorise Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 23 / 47
Enough causal states f g = = ⇒ = ∀ ρ causal . f g ρ ρ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 24 / 47
Second-order causal processes factorise ∃ Φ 1 , Φ 2 causal . ∀ Φ causal . Φ 2 = ⇒ = = w w Φ Φ 1 Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 25 / 47
Theorem In a pre-causal category, one-way signalling processes factorise: ∃ Φ 1 , Φ 2 causal . ∃ Φ ′ causal . = ⇒ Φ 2 = = Φ Φ Φ ′ Φ 1 Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 26 / 47
Proof. Treat Φ as a second-order process by bending wires. Then for any causal Ψ, we have: Φ Φ ′ Ψ = = = Φ ′ Ψ Ψ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 27 / 47
Proof. Treat Φ as a second-order process by bending wires. Then for any causal Ψ, we have: Φ Φ ′ Ψ = = = Φ ′ Ψ Ψ So Φ is second-order causal. By ( C4 ): Φ Φ 2 = Φ 1 Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 27 / 47
Proof. Treat Φ as a second-order process by bending wires. Then for any causal Ψ, we have: Φ Φ ′ Ψ = = = Φ ′ Ψ Ψ So Φ is second-order causal. By ( C4 ): Φ 2 Φ Φ 2 = = ⇒ = Φ Φ ′ 1 Φ 1 Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 27 / 47
Theorem (No time-travel) No non-trivial system A in a precausal category C admits time travel. That is, if there exist systems B and C such that: A C C = ⇒ causal causal Φ Φ A A B B then A ∼ = I. Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 28 / 47
Proof. For any causal process Ψ and causal state : A C A C := Φ Ψ A B A B is causal. Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 29 / 47
Proof. For any causal process Ψ and causal state : A C A C := Φ Ψ A B A B is causal.So: C = = = 1 Ψ Φ A A B B Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 29 / 47
Proof. For any causal process Ψ and causal state : A C A C := Φ Ψ A B A B is causal.So: C = = = 1 Ψ Φ A A B B Applying ( C4 ): A ρ A = = ⇒ = A A A ρ A for some ρ causal. Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 29 / 47
Proof. For any causal process Ψ and causal state : A C A C := Φ Ψ A B A B is causal.So: C = = = 1 Ψ Φ A A B B Applying ( C4 ): A ρ A = = ⇒ = A A A ρ A for some ρ causal.So ρ ◦ = 1 A Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 29 / 47
Proof. For any causal process Ψ and causal state : A C A C := Φ Ψ A B A B is causal.So: C = = = 1 Ψ Φ A A B B Applying ( C4 ): A ρ A = = ⇒ = A A A ρ A for some ρ causal.So ρ ◦ = 1 A and ◦ ρ = 1 I is causality. Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 29 / 47
Causal states A process is causal, a.k.a. first order causal , if and only if it preserves the set of causal states: = ⇒ f causal causal ρ ρ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 30 / 47
Causal states A process is causal, a.k.a. first order causal , if and only if it preserves the set of causal states: = ⇒ f causal causal ρ ρ That is, it preserves: � � � � c = ρ : A = 1 ⊆ C ( I , A ) ρ � � Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 30 / 47
Causal states A process is causal, a.k.a. first order causal , if and only if it preserves the set of causal states: = ⇒ f causal causal ρ ρ That is, it preserves: � � � � c = ρ : A = 1 ⊆ C ( I , A ) ρ � � Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 30 / 47
Causal states A process is causal, a.k.a. first order causal , if and only if it preserves the set of causal states: = ⇒ f causal causal ρ ρ That is, it preserves: � � � � c = ρ : A = 1 ⊆ C ( I , A ) ρ � � We define Caus [ C ] by equipping each object with a generalisation of the set c , and requiring processes to preserve it. Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 30 / 47
Duals and closure Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 31 / 47
Duals and closure Note any set of states c ⊆ C ( I , A ) admits a dual , which is a set of effects: � � � π c ∗ := � π : A ∗ � ∀ ρ ∈ c . = 1 � ρ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 31 / 47
Duals and closure Note any set of states c ⊆ C ( I , A ) admits a dual , which is a set of effects: � � � π c ∗ := � π : A ∗ � ∀ ρ ∈ c . = 1 � ρ The double-dual c ∗∗ is a set of states again. Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 31 / 47
Duals and closure Note any set of states c ⊆ C ( I , A ) admits a dual , which is a set of effects: � � � π c ∗ := � π : A ∗ � ∀ ρ ∈ c . = 1 � ρ The double-dual c ∗∗ is a set of states again. Definition A set of states c ⊆ C ( I , A ) is closed if c = c ∗∗ . Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 31 / 47
Flatness If c is the set of causal states, discarding ∈ c ∗ , and up to some rescaling, discarding-transpose: 1 D i.e. the maximally mixed state ∈ c . Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 32 / 47
Flatness If c is the set of causal states, discarding ∈ c ∗ , and up to some rescaling, discarding-transpose: 1 D i.e. the maximally mixed state ∈ c . We make this symmetric c ↔ c ∗ , and call this propery flatness: Definition A set of states c ⊆ C ( I , A ) is flat if there exist invertible numbers λ, µ such that: ∈ c ∗ λ ∈ c µ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 32 / 47
The main definition Definition For a precausal category C , the category Caus [ C ] has as objects pairs: ❆ := ( A , c ❆ ⊆ C ( I , A )) where c ❆ is closed and flat. A morphism f : ❆ → ❇ is a morphism f : A → B in C such that: ρ ∈ c ❆ = ⇒ f ◦ ρ ∈ c ❇ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 33 / 47
The main theorem Theorem Caus [ C ] is a ∗ -autonomous category, where: ❆ ⊗ ❇ := ( A ⊗ B , ( c ❆ ⊗ c ❇ ) ∗∗ ) ■ := ( I , { 1 I } ) ❆ ∗ := ( A ∗ , c ∗ ❆ ) Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 34 / 47
❆ ❇ ❆ ❇ Connectives One connective ⊗ becomes 3 interrelated ones: ❆ ⊗ ❇ ❆ ` ❇ := ( ❆ ∗ ⊗ ❇ ∗ ) ∗ ❆ ⊸ ❇ := ❆ ∗ ` ❇ ∼ = ( ❆ ⊗ ❇ ∗ ) ∗ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 35 / 47
❆ ❇ ❆ ❇ Connectives One connective ⊗ becomes 3 interrelated ones: ❆ ⊗ ❇ ❆ ` ❇ := ( ❆ ∗ ⊗ ❇ ∗ ) ∗ ❆ ⊸ ❇ := ❆ ∗ ` ❇ ∼ = ( ❆ ⊗ ❇ ∗ ) ∗ • ⊗ is the smallest joint state space that contains all product states Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 35 / 47
Connectives One connective ⊗ becomes 3 interrelated ones: ❆ ⊗ ❇ ❆ ` ❇ := ( ❆ ∗ ⊗ ❇ ∗ ) ∗ ❆ ⊸ ❇ := ❆ ∗ ` ❇ ∼ = ( ❆ ⊗ ❇ ∗ ) ∗ • ⊗ is the smallest joint state space that contains all product states • ` is the biggest joint state space normalised on all product effects: � π ξ � c ❆ ` ❇ = ρ : A ⊗ B = 1 � ∀ π ∈ c ∗ ❆ , ξ ∈ c ∗ ❇ . � ρ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 35 / 47
Connectives One connective ⊗ becomes 3 interrelated ones: ❆ ⊗ ❇ ❆ ` ❇ := ( ❆ ∗ ⊗ ❇ ∗ ) ∗ ❆ ⊸ ❇ := ❆ ∗ ` ❇ ∼ = ( ❆ ⊗ ❇ ∗ ) ∗ • ⊗ is the smallest joint state space that contains all product states • ` is the biggest joint state space normalised on all product effects: � π ξ � c ❆ ` ❇ = ρ : A ⊗ B = 1 � ∀ π ∈ c ∗ ❆ , ξ ∈ c ∗ ❇ . � ρ • ⊸ is the space of causal-state-preserving maps Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 35 / 47
❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { c ❆ ⊗ ❇ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { c ❆ ⊗ ❇ := ( c ❆ ⊗ c ❇ ) ∗∗ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { c ❆ ⊗ ❇ := ( c ❆ ⊗ c ❇ ) ∗∗ = ( ) ∗ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { c ❆ ⊗ ❇ := ( c ❆ ⊗ c ❇ ) ∗∗ = ( ) ∗ = all causal states Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
❆ ❇ ❆ ❇ ❆ ❇ Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { c ❆ ⊗ ❇ := ( c ❆ ⊗ c ❇ ) ∗∗ = ( ) ∗ = all causal states c ❆ ` ❇ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
❆ ❇ ❆ ❇ Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { c ❆ ⊗ ❇ := ( c ❆ ⊗ c ❇ ) ∗∗ = ( ) ∗ = all causal states � π ξ � c ❆ ` ❇ := ρ : A ⊗ B = 1 � ∀ π ∈ c ∗ ❆ , ξ ∈ c ∗ ❇ . � ρ Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
❆ ❇ ❆ ❇ Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { c ❆ ⊗ ❇ := ( c ❆ ⊗ c ❇ ) ∗∗ = ( ) ∗ = all causal states � � c ❆ ` ❇ := ρ : A ⊗ B = 1 � ρ � Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
❆ ❇ ❆ ❇ Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { c ❆ ⊗ ❇ := ( c ❆ ⊗ c ❇ ) ∗∗ = ( ) ∗ = all causal states � � c ❆ ` ❇ := ρ : A ⊗ B = 1 � ρ � Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
❆ ❇ ❆ ❇ Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { c ❆ ⊗ ❇ := ( c ❆ ⊗ c ❇ ) ∗∗ = ( ) ∗ = all causal states � � c ❆ ` ❇ := ρ : A ⊗ B = 1 = all causal states � ρ � Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
Example: first-order systems } ∗ ) First order := systems of the form ❆ = ( A , { c ❆ ⊗ ❇ := ( c ❆ ⊗ c ❇ ) ∗∗ = ( ) ∗ = all causal states � � c ❆ ` ❇ := ρ : A ⊗ B = 1 = all causal states � ρ � Theorem For first order systems, ❆ ⊗ ❇ ∼ = ❆ ` ❇ . Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 36 / 47
❆ ❆ ❇ ❇ ❆ ❆ ❇ ❇ ❆ ❆ ❇ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❆ ❇ ❇ When ⊗ � = ` Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 37 / 47
❆ ❆ ❇ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❆ ❇ ❇ When ⊗ � = ` For f.o. ❆ , ❆ ′ , ❇ , ❇ ′ : ( ❆ ⊸ ❆ ′ ) ` ( ❇ ⊸ ❇ ′ ) Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 37 / 47
❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❇ ❆ ❆ ❇ ❇ When ⊗ � = ` For f.o. ❆ , ❆ ′ , ❇ , ❇ ′ : ❆ ∗ ` ❆ ′ ` ❇ ∗ ` ❇ ′ ∼ ( ❆ ⊸ ❆ ′ ) ` ( ❇ ⊸ ❇ ′ ) = Aleks Kissinger and Sander Uijlen A categorical semantics for causal structure December 8, 2019 37 / 47
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