This is the fourth post in the Cartesian frames sequence. Read the first post here.
Previously, we defined controllables as the sets of possible worlds an agent can both ensure and prevent, and we defined observables as the sets of possible worlds such that the agent can implement all conditional policies.
Now that we have built up more language, we can redefine controllables and observables more categorically.
1. Controllables
1.1. Ensurables and Preventables
The categorical definition of ensurables is very simple.
Definition:Ensure(C) is the set of all S⊆W such that there exists a morphism ϕ:1S→C.
As an example, let C0=(A,E,⋅). Recall that 1S=({b},S,⋆), where b⋆s=s for all s∈S. E.g., if S0={w0,w3,w4}, then
1S0=w0w3w4b(w0w3w4).
If there is a morphism (g,h) from 1S0 to C0, this means that:
There is a function g from 1S0's agent {b} to C0's agent A, i.e., a function that always outputs a specific a∈A.
There is a function h from C0's environment E to 1S0's environment S0.
The specific a∈A picked out by g exactly implements that function h:E→S0.
That function h:E→S0 is exactly like the function you get by looking at that row, so a morphism (g,h):1S0→C0 is like a row in C0 that is entirely contained in S0. If there are multiple such rows, then there will be multiple distinct morphisms 1S0→C0 picking out different a∈A.
In "Biextensional Equivalence," we noted that 1S is like a passive observer who has a promise from the environment that the world will be in S. The existence of a morphism 1S→C means that there's an interface that allows a powerless bystander who has been promised S to play C's game. Since Agent(1S) only has one option, this interface must send that one option to some option for C's agent that is compatible with this promise.
Proof: Let C=(A,E,⋅) and let 1S=({b},S,⋆), where b⋆s=s for all s∈S. First, assume there exists a morphism (g,h):1S→C. Here, g:{b}→A and h:E→S. Consider the element g(b)∈A. It suffices to show that g(b)⋅e∈S for all e∈E. Indeed, g(b)⋅e=b⋆h(e)∈S.
Conversely, assume that there exists an a∈A, such that a⋅e∈S for all e∈E. Then, there is a morphism (g,h):1S→C given by g(b)=a, and h(e)=a⋅e. This is a morphism because
g(b)⋅e=a⋅e=h(e)=b⋆h(e)
for all b∈{b} and e∈E. □
Definition: Prevent(C)is the set of all S⊆W such that there exists a morphism ϕ1:1W∖S→C.
Proof: This follows from the proof for Ensure(C), substituting W∖S for S. □
Our categorical definition gives us a bunch of facts about how ensurability interacts with various operations on Cartesian frames. First, ensurability is monotonic in the existence of morphisms.
Claim: If there exists a morphism ϕ:C→D, then Ensure(C)⊆Ensure(D).
Proof: If S∈Ensure(C), there exists a morphism ψ:1S→C, so we have ϕ∘ψ:1S→D, so S∈Ensure(D). □
This fact justifies our interpretation of the existence of a morphism from C to D as saying that "D is at least as strong as C."
We also have that ensurables interact very strongly with sums and products. The ensurables of a product are the intersection of the original two agents' ensurables, and the ensurables of a sum are (usually) the union of the original two agents' ensurables.
This makes sense when we think of C⊕D as "there are two games, and the agent gets to choose which one we play," and C&D as "there are two games, and the environment gets to choose which one we play." The agent of C⊕D can make sure something happens if either C or D's agent could, whereas the agent of C&D can only ensure things that are ensurable across both games.
Claim:Ensure(C&D)=Ensure(C)∩Ensure(D).
Proof: Since & is a categorical product, if there exists a morphism from 1S to C and a morphism from 1S to D, there must exist a morphism from 1S to C&D. Thus Ensure(C&D)⊇Ensure(C)∩Ensure(D). Conversely, since & is a categorical product, there exist projection morphisms from C&D to C and from C&D to D, so Ensure(C&D)⊆Ensure(C)∩Ensure(D). □
This is the fourth post in the Cartesian frames sequence. Read the first post here.
Previously, we defined controllables as the sets of possible worlds an agent can both ensure and prevent, and we defined observables as the sets of possible worlds such that the agent can implement all conditional policies.
Now that we have built up more language, we can redefine controllables and observables more categorically.
1. Controllables
1.1. Ensurables and Preventables
The categorical definition of ensurables is very simple.
Definition: Ensure(C) is the set of all S⊆W such that there exists a morphism ϕ:1S→C.
As an example, let C0=(A,E,⋅). Recall that 1S=({b},S,⋆), where b⋆s=s for all s∈S. E.g., if S0={w0,w3,w4}, then
1S0=w0 w3 w4b(w0 w3 w4).
If there is a morphism (g,h) from 1S0 to C0, this means that:
That function h:E→S0 is exactly like the function you get by looking at that row, so a morphism (g,h):1S0→C0 is like a row in C0 that is entirely contained in S0. If there are multiple such rows, then there will be multiple distinct morphisms 1S0→C0 picking out different a∈A.
In "Biextensional Equivalence," we noted that 1S is like a passive observer who has a promise from the environment that the world will be in S. The existence of a morphism 1S→C means that there's an interface that allows a powerless bystander who has been promised S to play C's game. Since Agent(1S) only has one option, this interface must send that one option to some option for C's agent that is compatible with this promise.
Claim: This definition is equivalent to the one in "Introduction to Cartesian Frames": Ensure(C)={S⊆W | ∃a∈A,∀e∈E,a⋅e∈S}.
Proof: Let C=(A,E,⋅) and let 1S=({b},S,⋆), where b⋆s=s for all s∈S. First, assume there exists a morphism (g,h):1S→C. Here, g:{b}→A and h:E→S. Consider the element g(b)∈A. It suffices to show that g(b)⋅e∈S for all e∈E. Indeed, g(b)⋅e=b⋆h(e)∈S.
Conversely, assume that there exists an a∈A, such that a⋅e∈S for all e∈E. Then, there is a morphism (g,h):1S→C given by g(b)=a, and h(e)=a⋅e. This is a morphism because
g(b)⋅e=a⋅e=h(e)=b⋆h(e)for all b∈{b} and e∈E. □
Definition: Prevent(C) is the set of all S⊆W such that there exists a morphism ϕ1:1W∖S→C.
Claim: This definition is equivalent to the one in "Introduction to Cartesian Frames": Prevent(C)={S⊆W | ∃a∈A, ∀e∈E, a⋅e∉S}.
Proof: This follows from the proof for Ensure(C), substituting W∖S for S. □
Our categorical definition gives us a bunch of facts about how ensurability interacts with various operations on Cartesian frames. First, ensurability is monotonic in the existence of morphisms.
Claim: If there exists a morphism ϕ:C→D, then Ensure(C)⊆Ensure(D).
Proof: If S∈Ensure(C), there exists a morphism ψ:1S→C, so we have ϕ∘ψ:1S→D, so S∈Ensure(D). □
This fact justifies our interpretation of the existence of a morphism from C to D as saying that "D is at least as strong as C."
We also have that ensurables interact very strongly with sums and products. The ensurables of a product are the intersection of the original two agents' ensurables, and the ensurables of a sum are (usually) the union of the original two agents' ensurables.
This makes sense when we think of C⊕D as "there are two games, and the agent gets to choose which one we play," and C&D as "there are two games, and the environment gets to choose which one we play." The agent of C⊕D can make sure something happens if either C or D's agent could, whereas the agent of C&D can only ensure things that are ensurable across both games.
Claim: Ensure(C&D)=Ensure(C)∩Ensure(D).
Proof: Since & is a categorical product, if there exists a morphism from 1S to C and a morphism from 1S to D, there must exist a morphism from 1S to C&D. Thus Ensure(C&D)⊇Ensure(C)∩Ensure(D). Conversely, since & is a categorical product, there exist projection morphisms from C&D to C and from C&D to D, so Ensure(C&D)⊆Ensure(C)∩Ensure(D). □
Claim: If C≠null and D≠null, then Ensure