Functors and Coarse Worlds

This is the fifth post in the Cartesian frames sequence. Read the first post here.

Up until this point, we have only been working with Cartesian frames over a fixed world $W$ . Now, we are going to start talking about Cartesian frames over different worlds.

1. Functors from Functions Between Worlds

In the Cartesian frames framework, a world is a set of possible worlds $w$ that can all potentially occur in the same frame.

I find it useful to think about “different worlds” $W$ and $V$ in the case where $W$ and $V$ are different world models that carve up a situation in two different ways. $W$ might be a refined world model, one that describes a situation in more detail; while $V$ is a coarser model of the same situation that elides some distinctions in $W$ .

Returning to an example from “Biextensional Equivalence,” $W = {w_{0}, w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{6}, w_{7}}$ could be a world model that includes details about what the agent is thinking ( $G$ for a thought about the color green, $R$ for red), as shown in

$C_{0} = \begin{matrix} \begin{matrix} S & B \end{matrix} \begin{matrix} GH GW RH RW \end{matrix} & ⎛ ⎜ ⎜ ⎜ ⎝ \begin{matrix} w_{0} & w_{1} w_{2} & w_{3} w_{4} & w_{5} w_{6} & w_{7} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎠ \end{matrix}$ ,

while $V = {w_{8}, w_{9}, w_{10}, w_{11}}$ could be a world model that leaves out this information, representing the same real-world situation with the frame

$C_{1} = \begin{matrix} \begin{matrix} S & B \end{matrix} \begin{matrix} GH GW RH RW \end{matrix} & ⎛ ⎜ ⎜ ⎜ ⎝ \begin{matrix} w_{8} & w_{9} w_{10} & w_{11} w_{8} & w_{9} w_{10} & w_{11} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎠ \end{matrix}$ .

To move between frames like $C_{0}$ and $C_{1}$ and compare their properties, we will need a way to send agents and environments of frames defined over one world, to agents and environments of frames over an entirely different world. Functors will allow us to do this.

Definition: Given two sets $W$ and and $V$ , and a function $p : W \to V$ , let $p^{\circ} : Chu (W) \to Chu (V)$ denote the functor that sends the object $(A, E, \cdot) \in Chu (W)$ to the object $(A, E, ⋆) \in Chu (V)$ , where $a ⋆ e = p (a \cdot e)$ , and sends the morphism $(g, h)$ to the morphism with the same underlying functions, $(g, h)$ .

To visualize this functor, you can imagine $Chu (W)$ as a graph, with matrices as nodes (in the finite case) and arrows representing morphisms. $Chu (V)$ is another graph made of matrices and arrows. To move each frame $C$ from $Chu (W)$ to $Chu (V)$ , we use $p$ to entrywise replace the possible worlds in $C$ ’s matrix with elements of $V$ , without changing the functional properties of the rows and columns; and then we move all the arrows from $Chu (W)$ to $Chu (V)$ , which is possible because no functional properties of the original matrices were lost. (Frames and morphisms may or may not be added when we move to $Chu (V)$ .)

In the cases where we say ” $W$ is a refined version of $V$ ” or ” $V$ is a coarse version of $W$ ,” all we mean is that the function $p : W \to V$ is surjective.

Claim: $p^{\circ}$ is well-defined.

Proof: We need to show that $p^{\circ}$ actually sends objects and morphisms of $Chu (W)$ to objects and morphisms of $Chu (V)$ , and that it preserves identity morphisms and composition. $p^{\circ}$ clearly sends objects to objects. To see that $p^{\circ}$ sends morphisms to morphisms, observe that if $(g, h) : (A_{0}, E_{0}, \cdot_{0}) \to (A_{1}, E_{1}, \cdot_{1})$ , and $p^{\circ} (A_{i}, E_{i}, \cdot_{i}) = (A_{i}, E_{i}, ⋆_{i})$ , then for all $a \in A_{0}$ and $e \in E_{1}$ ,

\begin{matrix} g (a) ⋆_{1} e & = p (g (a) \cdot_{1} e) = p (a \cdot_{0} h (e)) = a ⋆_{0} h (e), \end{matrix}

so $p^{\circ} (g, h) = (g, h)$ is a morphism. It is clear that $p^{\circ}$ preserves identity and composition, since it has no effect on morphisms. $□$

We also have that $p^{\circ}$ preserves all of our additive operations.

Claim: $p^{\circ} (C \oplus D) = p^{\circ} (C) \oplus p^{\circ} (D)$ , $p^{\circ} (C & D) = p^{\circ} (C) & p^{\circ} (D)$ , $p^{\circ} (C^{*}) = p^{\circ} (C)^{*}$ , $p^{\circ} (0) = 0$ , $p^{\circ} (⊤) = ⊤$ , and $p^{\circ} (null) = null$ .

Proof: Trivial. $□$

Our new functor’s relationship with $1$ and $⊥$ is more interesting. In particular, we can define $1_{S}$ and $⊥_{S}$ from $1$ and $⊥$ using functors.

Claim: Let $S \subseteq W$ and let $ι : S \to W$ be the inclusion of $S$ in $W$ . Then $1_{S} = ι^{\circ} (1)$ and $⊥_{S} = ι^{\circ} (⊥)$ . (Here, the $1$ and $⊥$ are from $Chu (S)$ , not $Chu (W)$ .)

Proof: Trivial. $□$

This gives us a more categorical definition of $1_{S}$ and $⊥_{S}$ from $1$ and $⊥$ . We will give a more categorical definition of $1$ and $⊥$ later, when we talk about multiplicative operations.

$p^{\circ}$ also preserves biextensional equivalence in one direction. (Two equivalent frames in $W$ will always be equivalent in $V$ , but two inequivalent frames in $W$ won’t necessarily be inequivalent in $V$ .)

Claim: If $C ≃ D$ , then $p^{\circ} (C) ≃ p^{\circ} (D)$ .

Proof: Let $C = (A, E, \cdot)$ and let $D = (B, F, ⋆)$ . Let $(g_{0}, h_{0}) : C \to D$ and $(g_{1}, h_{1}) : D \to C$ compose to something homotopic to the identity in both orders. We want to show that $(g_{0}, h_{0}) : p^{\circ} (C) \to p^{\circ} (D)$ and $(g_{1}, h_{1}) : p^{\circ} (D) \to p^{\circ} (C)$ compose to something homotopic to the identity in both orders. Indeed $p (g_{1} (g_{0} (a)) \cdot e) = p (a \cdot e)$ for all $a \in A$ and $e \in E$ , and $p (g_{0} (g_{1} (b)) ⋆ f) = p (b ⋆ f)$ for all $b \in B$ and $f \in F$ . $□$

We also have that $p^{\circ}$ preserves what’s ensurable, where we transition from subsets of $W$ to subsets of $V$ in the obvious way.

Claim: Let $p : W \to V$ , and let $p (S) = {v \in V | \exists w \in S, p (w) = v}$ . If $S \in Ensure (C)$ , then $p (S) \in Ensure (p^{\circ} (C))$ .

Proof: Trivial from the original definition of ensurables. $□$

We also get a stronger result when dealing with subsets of $W$ and $V$ that correspond exactly.

Claim: Let $p : W \to V$ , and let $S \subseteq W$ and $T \subseteq V$ be such that for all $w \in W$ , we have $p (w) \in T$ if and only if $w \in S$ . Then $S \in Ensure (C)$ if and only if $T \in Ensure (p^{\circ} (C))$ , and $S \in Ctrl (C)$ if and only if $T \in Ctrl (p^{\circ} (C))$ .

Proof: Trivial from the original definitions of ensurables and controllables. $□$

The relationship between observability and functors is quite interesting. We will devote the next section to discussing this relationship and its philosophical consequences.

2. What’s Observable is Relative to a Coarse World Model

Since observability is not closed under supersets, we can only really hope to get a result for observables in the stronger case where $S \subseteq W$ and $T \subseteq V$ correspond exactly; but interestingly, even then, the preservation result for observables is only one-directional.

Claim: Let $p : W \to V$ and let $S \subseteq W$ and $T \subseteq V$ be such that for all $w \in W$ , we have $p (w) \in T$ if and only if $w \in S$ . Then if $S \in Obs (C)$ , then $T \in Obs (p^{\circ} (C))$ .

Proof: If $C ≃ C_{0} & C_{1}$ , with $Image (C_{0}) \subseteq S$ and $Image (C_{1}) \subseteq W ∖ S$ , then $p^{\circ} (C) ≃ p^{\circ} (C_{0}) & p^{\circ} (C_{1})$ , and $Image (p^{\circ} (C_{0})) = p (Image (C_{0})) \subseteq p (S) \subseteq T$ , while $Image (p^{\circ} (C_{1})) = p (Image (C_{1})) \subseteq p (W ∖ S) \subseteq V ∖ T$ . $□$

The most interesting thing here is that the converse is not also true. There are examples where $T \in Obs (p^{\circ} (C))$ , even though $S \notin Obs (C)$ .

When $p$ is surjective, we think of $V$ as a coarse world model that forgets some details from $W$ . Sometimes, an agent can be able to observe $S$ relative to a coarse description of the world, but not in the more refined description, even in cases where $S$ is definable in both the coarse and refined descriptions.

2.1. Example

Let us look at an example. In this example, the agent is an AI that will be given a number and asked whether it is prime or not. There are two possible environments $E = {Prime, Nonprime}$ .

The agent $A$ has six strategies:

It can compute whether the number is prime and answer correctly ( $AccurateHot$ ), or it can compute whether the number is prime and answer incorrectly $(InaccurateHot)$ . In both of these cases, it has to use a lot of energy and will become hotter.
It can also just not think and say that the number is prime $(PrimeCool)$ , or it can not think and say that the number is not prime ( $NonprimeCool$ ).
Finally, it can deliberately waste a lot of energy for no reason and say the number is prime or waste a lot of energy and say the number is not prime ( $PrimeHot$ and $NonprimeHot$ respectively).

Finally, $W = {PAH, PAC, PIH, PIC, NAH, NAC, NIH, NIC}$ , where the first letter indicates whether the AI was given a prime or nonprime number, the second letter indicates whether the AI’s answer was accurate or inaccurate, and the third letter indicates whether the AI is hot. The Cartesian frame, $C$ , looks like this.

$C = \begin{matrix} \begin{matrix} Prime & Nonprime \end{matrix} \begin{matrix} AccurateHot InaccurateHot PrimeCool NonprimeCool PrimeHot NonprimeHot \end{matrix} & ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} PAH & NAH PIH & NIH PAC & NIC PIC & NAC PAH & NIH PIH & NAH \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ \end{matrix}$

We will let $V$ be the coarse description of the world in which we only pay attention to the input/output behavior of the AI and ignore whether or not it becomes hot. $V = {PA, PI, NA, NI}$ , and we will let $p : W \to V$ be the function that deletes the third letter. This gives us the following for $p^{\circ} (C)$ .

$p^{\circ} (C) = \begin{matrix} \begin{matrix} Prime & Nonprime \end{matrix} \begin{matrix} AccurateHot InaccurateHot PrimeCool NonprimeCool PrimeHot NonprimeHot \end{matrix} & ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} PA & NA PI & NI PA & NI PI & NA PA & NI PI & NA \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ \end{matrix} ≃ \begin{matrix} \begin{matrix} Prime & Nonprime \end{matrix} \begin{matrix} Accurate Inaccurate Prime Nonprime \end{matrix} & ⎛ ⎜ ⎜ ⎜ ⎝ \begin{matrix} PA & NA PI & NI PA & NI PI & NA \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎠ \end{matrix}$

The important thing to notice here is that ${PA, PI} \in Obs (p^{\circ} (C))$ —when we ignore heat, the agent can base conditional strategies on whether the number is prime—but ${PAH, PAC, PIH, PIC} \notin Obs (C)$ .

In particular, $p^{\circ} (C) ≃ C_{0} & C_{1}$ , where

$C_{0} = \begin{matrix} \begin{matrix} Prime \end{matrix} \begin{matrix} Accurate Inaccurate \end{matrix} & (\begin{matrix} PA PI \end{matrix}) \end{matrix}$ and $C_{1} = \begin{matrix} \begin{matrix} Nonprime \end{matrix} \begin{matrix} Accurate Inaccurate \end{matrix} & (\begin{matrix} NA NI \end{matrix}) \end{matrix}$ ,

while it is easy to see that ${PAH, PAC, PIH, PIC} \notin Obs (C)$ , because there is no $a \in if ({PAH, PAC, PIH, PIC}, PrimeCool, NonprimeCool)$ .

2.2. Discussion

The above example illustrates something interesting about observables. It shows that what’s observable is not only a function of the observing agent and the thing that is observed. It is also a function of the level of description of the world!

This makes sense because we are thinking of observation as the ability to implement conditional policies. To implement a conditional policy is to be indistinguishable from the constant policy $a_{0}$ in worlds in $S$ and indistinguishable from the constant policy $a_{1}$ in worlds outside of $S$ . This indistinguishability makes observables relative to the level of description of the world.

There is something internal to the agent that is different between the world where it implements a conditional policy and the world where it implements a constant policy. However, when we talk of $S$ being an observable for the agent, we are working relative to a level of description that does not track that internal difference.

3. Functors from Cartesian Frames

When $p : W \to V$ is surjective, $p^{\circ}$ will send Cartesian frames over the more refined $W$ to Cartesian frames over the less refined $V$ . What if we want to go in the other direction?

While there is a unique function from less refined worlds to more refined worlds, there are many functions in the other direction. Luckily, we have an object that lets us deal with many functions at once.

Definition: Let $C = (V, E, \cdot)$ be a Cartesian frame over $W$ , with $Agent (C) = V$ . Then $C^{\circ} : Chu (V) \to Chu (W)$ is the functor that sends $(B, F, ⋆)$ to $(B, F \times E, ⋄)$ , where $b ⋄ (f, e) = (b ⋆ f) \cdot e$ , and sends the morphism $(g, h)$ to $(g, h^{'})$ , where $h^{'} (f, e) = (h (f), e)$ .

(Notice how this definition looks a bit like currying.)

Claim: $C^{\circ}$ is well-defined.

Proof: We need to show that $C^{\circ}$ actually sends objects and morphisms of $Chu (V)$ to objects and morphisms of $Chu (W)$ , and that it preserves identity morphisms and composition.

$C^{\circ}$ clearly sends objects to objects. To see that it sends morphisms to morphisms, let $(g, h) : (B_{0}, F_{0}, ⋆_{0}) \to (B_{1}, F_{1}, ⋆_{1})$ be a morphism in $Chu (V)$ , let $(B_{i}, F_{i} \times E, ⋄_{i}) = C^{\circ} (B_{i}, F_{i}, ⋆_{i})$ , and let $(g, h^{'}) = C^{\circ} (g, h)$ .

We want to show that $(g, h^{'}) : (B_{0}, F_{0} \times E, ⋄_{0}) \to (B_{1}, F_{1} \times E, ⋄_{1})$ is a morphism, which is true because

\begin{matrix} g (b) ⋄_{1} (f, e) & = (g (b) ⋆_{1} f) \cdot e = (b ⋆_{0} h (f)) \cdot e = b ⋄_{0} (h (f), e) = b ⋄_{0} h^{'} (f, e) \end{matrix}

for all $b \in B_{0}$ and $(f, e) \in F_{1} \times E$ . $C^{\circ}$ clearly preserves identity morphisms and composition. $□$

The coarse-to-refined functor $C^{\circ}$ preserves $&$ , $⊤$ , and $null$ , but not $\oplus$ , $0$ , or $-^{*}$ , which make sense, since $C^{\circ}$ is violating the symmetry between agent and environment.

Claim: $C^{\circ} (⊤) = ⊤$ , and $C^{\circ} (null) = null$ .

Proof: Trivial. $□$

Claim: $C^{\circ} (D_{0} & D_{1}) = C^{\circ} (D_{0}) & C^{\circ} (D_{1})$ .

Proof: Let $C = (V, E, \cdot)$ and let $D_{i} = (B_{i}, F_{i}, ⋆_{i})$ . We have that $C^{\circ} (D_{0} & D_{1}) = (B_{0} \times B_{1}, (F_{0} ⊔ F_{1}) \times E, ⋄)$ and $C^{\circ} (D_{0}) & C^{\circ} (D_{1}) = (B_{0} \times B_{1}, (F_{0} \times E) ⊔ (F_{1} \times E), ∙)$ . The agent and environment are the same, so we just need to check that $⋄ = ∙$ .

Take $(b_{0}, b_{1}) \in B_{0} \times B_{1}$ and $(f, e) \in (F_{0} ⊔ F_{1}) \times E = (F_{0} \times E) ⊔ (F_{1} \times E)$ . Without loss of generality, assume $f \in F_{0}$ . Observe that

\begin{matrix} (b_{0}, b_{1}) ⋄ (f, e) & = (b_{0} ⋆_{0} f) \cdot e = (b_{0}, b_{1}) ∙ (f, e) . \end{matrix}

$□$

One way to see that $C^{\circ}$ does not preserve $\oplus$ is to see that the environments are different, since $C^{\circ} (D_{0} \oplus D_{1})$ has one copy of $E$ in the environment, while $C^{\circ} (D_{0}) \oplus C^{\circ} (D_{1})$ has two copies.

We also have that $C^{\circ}$ preserves biextensional equivalence.

Claim: if $D_{0} ≃ D_{1}$ , then $C^{\circ} (D_{0}) ≃ C^{\circ} (D_{1})$ .

Proof: Let $D_{i} = (B_{i}, F_{i}, ⋆_{i})$ , and let $C^{\circ} (D_{i}) = (B_{i}, F_{i} \times E, ⋄_{i})$ . Let $(g_{0}, h_{0}) : D_{0} \to D_{1}$ and $(g_{1}, h_{1}) : D_{1} \to D_{0}$ compose to something homotopic to the identity in both orders. It suffices to show that $C^{\circ} (g_{1}, h_{1}) \circ C^{\circ} (g_{0}, h_{0})$ is homotopic to the identity on $C^{\circ} (D_{0})$ , since the other composition will be symmetric. Indeed

\begin{matrix} g_{1} (g_{0} (b)) ⋄_{0} (f, e) & = g_{1} (g_{0} (b)) ⋆_{0} f = b ⋆_{0} f = b ⋄_{0} (f, e) \end{matrix}

for all $b \in B_{0}$ , and $(f, e) \in F_{0} \times E$ . $□$

Before we talk about the relationship between functors from functions and functors from Cartesian frames, I want to pause to talk about how to view Cartesian frames as sets of functions.

4. Cartesian Frames as Sets of Functions

One way to view (some) Cartesian frames is as sets of functions.

Definition: Given a set $P$ of functions from $E$ to $W$ , let $CF (P)$ denote the Cartesian frame over $W$ given by $(P, E, \cdot)$ , where $p \cdot e = p (e)$ .

Claim: $CF (P)$ is well-defined.

Proof: Trivial. $□$

Not every Cartesian Frame is expressible this way: every Cartesian frame is biextensionally equivalent to a Cartesian frame with duplicate columns and rows, and these uncollapsed frames are excluded because sets do not allow multiplicity.

Claim: For every Cartesian frame $C$ over $W$ , there exists a set of functions $P : Env (C) \to W$ , such that $C ≃ CF (P)$ .

Proof: Take $C = (A, E, \cdot)$ , and take $P$ to be the set of all $p : E \to W$ such that there exists an $a \in A$ such that for all $e \in E$ , $p (e) = a \cdot e$ . Take $(g_{0}, h_{0}) : C \to CF (P)$ and $(g_{1}, h_{1}) : CF (P) \to C$ , given as follows: $h_{0} = h_{1}$ is the identity on $E$ , $g_{0} (a)$ is the function $e \mapsto a \cdot e$ , and $g_{1} (p)$ is some $a \in A$ such that $p (-) = a \cdot -$ . These are both clearly morphisms, and they compose to something homotopic to the identity, since $h_{0} \circ h_{1}$ and $h_{1} \circ h_{0}$ are both the identity. $□$

This give us an alternate definition of Cartesian frames up to biextensional equivalence. This almost gives a complete alternate definition of Cartesian frames; if we instead took $P$ to be a multiset, then we could identify the Cartesian frame $CF (P)$ with the multiset $P$ .

Note that this is not as symmetric as our original definition of Cartesian frames. The “sets of functions” approach here thinks of a Cartesian frame as a set of functions from the environment to the world, but we could instead think of it as a set of functions from the agent to the world.

Definition: Given a set $P$ of functions for $A$ to $W$ , let ${CF}^{*} (P)$ denote the Cartesian frame over $W$ given by $(A, P, \cdot)$ , where $a \cdot p = p (a)$ .

Claim: ${CF}^{*} (P) = (CF (P))^{*}$ .

Proof: Trivial. $□$

Thinking of Cartesian frames in this way is not particularly different from our original definition. It is just thinking about a function with two inputs as a parameterized function with one input and one parameter. However, this way of understanding Cartesian frames will allow us to more easily relate functors from functions to functors from Cartesian frames.

5. Relationship Between the Two Functor Definitions

Functors from functions are a special case of functors from Cartesian frames. Indeed, they correspond when $Env (C)$ is a singleton.

Claim: For any $p : V \to W$ , $p^{\circ} = ({CF}^{*} ({p}))^{\circ}$ . Conversely, if $C = (V, {e}, \cdot)$ is a Cartesian frame over $W$ with singleton environment, then $C^{\circ} = p^{\circ}$ , where $p (v) = v \cdot e$ .

Proof: Observe that ${CF}^{*} ({p}) = (V, {e}, \cdot)$ , where $v \cdot e = p (e)$ . That $p^{\circ} = ({CF}^{*} ({p}))^{\circ}$ is trivial from considering the definition of $C^{\circ}$ in the special case where $E$ is a singleton. $□$

However, we can do a lot more with functors from Cartesian frames. In the case where $p : W \to V$ is a surjection, $p^{\circ}$ shows how to send Cartesian frames over the more refined $W$ to the less refined $V$ . We want to go in the other direction using an inverse of $p$ .

Since $p$ is a surjection, it has a right inverse, but it might have many right inverses. If we want to go from Cartesian frames over $V$ to Cartesian frames over $W$ , we could pick any right inverse to $p$ , but since we have functors from Cartesian frames, we don’t have to.

Claim: For any surjective $p : W \to V$ , let $Q$ be the set of all $q : V \to W$ such that $p \circ q$ is the identity on $V$ . Then for any Cartesian frame $C$ over $V$ , $(p^{\circ} \circ ({CF}^{*} (Q))^{\circ}) (C) ≃ C$ . Thus $({CF}^{*} (Q))^{\circ}$ is right inverse to $p^{\circ}$ up to biextensional equivalence.

Proof: Let $C = (A, E, \cdot)$ . Then $({CF}^{*} (Q))^{\circ} (C) = (A, E \times Q, ⋆)$ , where $a ⋆ (e, q) = q (a \cdot e),$ and $(p^{\circ} \circ ({CF}^{*} (Q))^{\circ}) (C) = (A, E \times Q, ⋄),$ where

\begin{matrix} a ⋄ (e, q) & = p (q (a \cdot e)) = a \cdot e . \end{matrix}

(Viewed as a matrix, $(p^{\circ} \circ ({CF}^{*} (Q))^{\circ}) (C)$ is isomorphic to $C$ with $| Q |$ copies of each column.)

To explicitly see the homotopy equivalence, take $(g_{0}, h_{0}) : (A, E, \cdot) \to (A, E \times Q, ⋄)$ by $g_{0} (a) = a$ and $g_{1} (e, q) = e$ , and take $(g_{1}, h_{1}) : (A, E \times Q, ⋄) \to (A, E, \cdot)$ by $g_{1} (a) = a$ and $h_{1} (e) = (e, q)$ for some fixed $q \in Q$ . These are clearly morphisms and clearly compose to something homotopic to the identity in both orders, since the $g_{i}$ are the identity. Note that we used the surjectivity of $p$ when we said “for some fixed $q \in Q$ ,” since the surjectivity of $p$ is what makes $Q$ nonempty. $□$

Functors from Cartesian frames will prove useful in the next section, when we finally introduce the concept of subagent.