Cartesian Frames Definitions

This is a list of the main definitions from Scott Garrabrant’s Cartesian Frames sequence. (I’ll update it as more posts come out.)

1. Small Cartesian Frames

Let $W = {w_{0}, w_{1}}$ for the matrix visualizations below. Let $C$ be an arbitrary Cartesian frame.

	visualization	definition	notes
$0$	$\begin{matrix} \begin{matrix} e \end{matrix} \begin{matrix} \end{matrix} & (\begin{matrix} \end{matrix}) \end{matrix}$	$0 = ({}, {e}, \cdot)$ , where $Agent (0)$ is empty, $Env (0) = {e}$ is any singleton set, and $Eval (0)$ is trivial.	$⊥_{{}}$ . Initial. Identity of sum ( $\oplus$ ).
$⊤$	$\begin{matrix} \begin{matrix} \end{matrix} \begin{matrix} a \end{matrix} & (\begin{matrix} \end{matrix}) \end{matrix}$	$⊤ = ({a}, {}, \cdot)$ , where $Agent (⊤)$ is any singleton set, $Env (⊤)$ is empty, and $Eval (⊤)$ is trivial.	$1_{{}}$ . Terminal. $⊤ ◃ C$ . Identity of product ( $&$ ).
$1$	$\begin{matrix} \begin{matrix} w_{0} & w_{1} \end{matrix} \begin{matrix} a \end{matrix} & (\begin{matrix} w_{0} & w_{1} \end{matrix}) \end{matrix}$	$1_{S} = ({a}, S, ⋆)$ , where $S \subseteq W$ and $a ⋆ s = s$ for all $s \in S$ . $1$ is the frame $1_{W}$ .	Identity of tensor ( $\otimes$ ).
$⊥$	$\begin{matrix} \begin{matrix} e \end{matrix} \begin{matrix} w_{0} w_{1} \end{matrix} & (\begin{matrix} w_{0} w_{1} \end{matrix}) \end{matrix}$	$⊥_{S} = (S, {e}, ⋆)$ , where $S \subseteq W$ and $s ⋆ e = s$ for all $s \in S$ . $⊥$ is the frame $⊥_{W}$ .	$C ◃ ⊥$ . Identity of par ( $⅋$ ).
$null$	$\begin{matrix} \begin{matrix} \end{matrix} \begin{matrix} \end{matrix} & (\begin{matrix} \end{matrix}) \end{matrix}$	$null = ({}, {}, \cdot)$ , with empty agent, environment, and evaluation function.

2. Binary Operations

Sum. For Cartesian frames $C = (A, E, \cdot)$ and $D = (B, F, ⋆)$ over $W$ , $C \oplus D$ is the Cartesian frame $(A ⊔ B, E \times F, ⋄)$ , where $a ⋄ (e, f) = a \cdot e$ if $a \in A$ , and $a ⋄ (e, f) = a ⋆ f$ if $a \in B$ .

Product. For Cartesian frames $C = (A, E, \cdot)$ and $D = (B, F, ⋆)$ over $W$ , $C & D$ is the Cartesian frame $(A \times B, E ⊔ F, ⋄)$ , where $(a, b) ⋄ e = a \cdot e$ if $e \in E$ , and $(a, b) ⋄ e = b ⋆ e$ if $e \in F$ .

Tensor. Let $C = (A, E, \cdot)$ and $D = (B, F, ⋆)$ be Cartesian frames over $W$ . The tensor product of $C$ and $D$ , written $C \otimes D$ , is given by $C \otimes D = (A \times B, hom (C, D^{*}), ⋄)$ , where $hom (C, D^{*})$ is the set of morphisms $(g, h) : C \to D^{*}$ (i.e., the set of all pairs $(g : A \to F, h : B \to E)$ such that $b ⋆ g (a) = a \cdot h (b)$ for all $a \in A$ , $b \in B$ ), and $⋄$ is given by $(a, b) ⋄ (g, h) = b ⋆ g (a) = a \cdot h (b)$ .

Par. Let $C = (A, E, \cdot)$ and $D = (B, F, ⋆)$ be Cartesian frames over $W$ . $C ⅋ D = (hom (C^{*}, D), E \times F, ⋄)$ , where $(g, h) ⋄ (e, f) = g (e) ⋆ f = h (f) \cdot e$ .

Lollipop. Given two Cartesian frames over $W$ , $C = (A, E, \cdot)$ and $D = (B, F, ⋆)$ , we let $C ⊸ D$ denote the Cartesian frame $C ⊸ D = (hom (C, D), A \times F, ⋄)$ , where $⋄$ is given by $(g, h) ⋄ (a, f) = g (a) ⋆ f = a \cdot h (f)$ .

3. Frames, Morphisms, and Equivalence Relations

Cartesian frame. A Cartesian frame $C$ over a set $W$ is a triple $(A, E, \cdot)$ , where $A$ and $E$ are sets and $\cdot : A \times E \to W$ . If $C = (A, E, \cdot)$ is a Cartesian frame over $W$ , we say $Agent (C) = A$ , $Env (C) = E$ , $World (C) = W$ , and $Eval (C) = \cdot$ .

Environment subset. Given a Cartesian frame $C = (A, E, \cdot)$ over $W$ , and a subset $S$ of $W$ , let $E_{S}$ denote the subset ${e \in E | \forall a \in A, e \cdot a \in S}$ .

Cartesian frame image. $Image (C) = {w \in W | \exists a \in A, \exists e \in E s.t. a \cdot e = w}$ .

Chu category. $Chu (W)$ is the category whose objects are Cartesian frames over $W$ , whose morphisms from $C = (A, E, \cdot)$ to $D = (B, F, ⋆)$ are pairs of functions $(g : A \to B, h : F \to E)$ , such that $a \cdot h (f) = g (a) ⋆ f$ for all $a \in A$ and $f \in F$ , and whose composition of morphisms is given by $(g_{1}, h_{1}) \circ (g_{0}, h_{0}) = (g_{1} \circ g_{0}, h_{0} \circ h_{1})$ .

Isomorphism. A morphism $(g, h) : C \to D$ is an isomorphism if both $g$ and $h$ are bijective. If there is an isomorphism between $C$ and $D$ , we say $C ≅ D$ .

Homotopic. Two morphisms $(g_{0}, h_{0}), (g_{1}, h_{1}) : C \to D$ with the same source and target are called homotopic if $(g_{0}, h_{1})$ is also a morphism.

Homotopy equivalence / biextensional equivalence. $C$ is homotopy equivalent (or biextensionally equivalent) to $D$ , written $C ≃ D$ , if there exists a pair of morphisms $ϕ : C \to D$ and $ψ : D \to C$ such that $ψ \circ ϕ$ is homotopic to the identity on $C$ and $ϕ \circ ψ$ is homotopic to the identity on $D$ .

Sub-sum. Let $C = (A, E, \cdot)$ , and let $D = (B, F, ⋆)$ . A sub-sum of C and D is a Cartesian frame of the form $(A ⊔ B, X, ⋄)$ , where $X \subseteq Env (C \oplus D)$ and $⋄$ is $Eval (C \oplus D)$ restricted to $(A ⊔ B) \times X$ , such that $C ≃ (A, X, ⋄_{C})$ and $D ≃ (B, X, ⋄_{D})$ , where $⋄_{C}$ is $⋄$ restricted to $A \times X$ and $⋄_{D}$ is $⋄$ restricted to $B \times X$ . Let $C ⊞ D$ denote the set of all sub-sums of $C$ and $D$ .

Sub-tensor. Let $C = (A, E, \cdot)$ , and let $D = (B, F, ⋆)$ . A sub-tensor of $C$ and $D$ is a Cartesian frame of the form $(A \times B, X, ∙)$ , where $X \subseteq Env (C \otimes D)$ and $∙$ is $Eval (C \otimes D)$ restricted to $(A \times B) \times X$ , such that $C ≃ (A, B \times X, ∙_{C})$ and $D ≃ (B, A \times X, ∙_{D})$ , where $∙_{C}$ and $∙_{D}$ are given by $a ∙_{C} (b, x) = (a, b) ∙ x$ and $b ∙_{D} (a, x) = (a, b) ∙ x$ . Let $C ⊠ D$ denote the set of all sub-tensors of $C$ and $D$ .

4. Functors

Functions between worlds. Given a Cartesian frame $C = (A, E, \cdot)$ over $W$ , and a function $f : W \to V$ , let $f^{\circ} (C)$ denote the Cartesian frame over $V$ , $f^{\circ} (C) = (A, E, ⋆)$ , where $a ⋆ e = f (a \cdot e)$ .

Dual. Let $-^{*} : Chu (W) \to Chu (W)^{op}$ be the functor given by $(A, E, \cdot)^{*} = (E, A, ⋆)$ , where $e ⋆ a = a \cdot e$ , and $(g, h)^{*} = (h, g)$ .

Functor (from functions between worlds). 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, $(g, h)$ .

Functor (from Cartesian frames). 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 $h^{'} (f, e) = (h (f), e)$ .

5. Subagents

Subagent (categorical definition). Let $C$ and $D$ be Cartesian frames over $W$ . We say that $C$ is a subagent of $D$ , written $C ◃ D$ , if for every morphism $ϕ : C \to ⊥$ there exists a pair of morphisms $ϕ_{0} : C \to D$ and $ϕ_{1} : D \to ⊥$ such that $ϕ = ϕ_{1} \circ ϕ_{0}$ .

Subagent (currying definition). Let $C$ and $D$ be Cartesian frames over $W$ . We say that $C ◃ D$ if there exists a Cartesian frame $Z$ over $Agent (D)$ such that $C ≃ D^{\circ} (Z)$ .

Subagent (covering definition). Let $C = (A, E, \cdot)$ and $D = (B, F, ⋆)$ be Cartesian frames over $W$ . We say that $C ◃ D$ if for all $e \in E$ , there exists an $f \in F$ and a $(g, h) : C \to D$ such that $e = h (f)$ .

Sub-environment. We say $C$ is a sub-environment of $D$ , written $C ◃^{*} D$ , if $D^{*} ◃ C^{*}$ .

5.1. Additive and Multiplicative Subagents

Additive subagent (sub-sum definition). $C$ is an additive subagent of $D$ , written $C ◃_{+} D$ , if there exists a $C^{'}$ and a $D^{'} ≃ D$ with $D^{'} \in C ⊞ C^{'}$ .

Additive subagent (brother definition). $C^{'}$ is called a brother to $C$ in $D$ if $D ≃ D^{'}$ for some $D^{'} \in C ⊞ C^{'}$ . We say $C ◃_{+} D$ if $C$ has a brother in $D$ .

Additive subagent (committing definition). Given Cartesian frames $C$ and $D$ over $W$ , we say $C ◃_{+} D$ if there exist three sets $X$ , $Y$ , and $Z$ , with $X \subseteq Y$ , and a function $f : Y \times Z \to W$ such that $C ≃ (X, Z, ⋄)$ and $D ≃ (Y, Z, ∙)$ , where $⋄$ and $∙$ are given by $x ⋄ z = f (x, z)$ and $y ∙ z = f (y, z)$ .

Additive subagent (currying definition). We say $C ◃_{+} D$ if there exists a Cartesian frame $M$ over $Agent (D)$ with $| Env (M) | = 1$ , such that $C ≃ D^{\circ} (M)$ .

Additive subagent (categorical definition). We say $C ◃_{+} D$ if there exists a single morphism $ϕ_{0} : C \to D$ such that for every morphism $ϕ : C \to ⊥$ there exists a morphism $ϕ_{1} : D \to ⊥$ such that $ϕ$ is homotopic to $ϕ_{1} \circ ϕ_{0}$ .

Multiplicative subagent (sub-tensor definition). $C$ is a multiplicative subagent of $D$ , written $C ◃_{\times} D$ , if there exists a $C^{'}$ and $D^{'} ≃ D$ with $D^{'} \in C ⊠ C^{'}$ .

Multiplicative subagent (sister definition). $C^{'}$ is called a sister to $C$ in $D$ if $D ≃ D^{'}$ for some $D^{'} \in C ⊠ C^{'}$ . We say $C ◃_{\times} D$ if $C$ has a sister in $D$ .

Multiplicative subagent (externalizing definition). Given Cartesian frames $C$ and $D$ over $W$ , we say $C ◃_{\times} D$ if there exist three sets $X$ , $Y$ , and $Z$ , and a function $f : X \times Y \times Z \to W$ such that $C ≃ (X, Y \times Z, ⋄)$ and $D ≃ (X \times Y, Z, ∙)$ , where $⋄$ and $∙$ are given by $x ⋄ (y, z) = f (x, y, z)$ and $(x, y) ∙ z = f (x, y, z)$ .

Multiplicative subagent (currying definition). We say $C ◃_{\times} D$ if there exists a Cartesian frame $M$ over $Agent (D)$ with $Image (M) = Agent (D)$ , such that $C ≃ D^{\circ} (M)$ .

Multiplicative subagent (categorical definition). We say $C ◃_{\times} D$ if for every morphism $ϕ : C \to ⊥$ , there exist morphisms $ϕ_{0} : C \to D$ and $ϕ_{1} : D \to ⊥$ such that $ϕ ≅ ϕ_{1} \circ ϕ_{0}$ , and for every morphism $ψ : 1 \to D$ , there exist morphisms $ψ_{0} : 1 \to C$ and $ψ_{1} : C \to D$ such that $ψ ≅ ψ_{1} \circ ψ_{0}$ .

Multiplicative subagent (sub-environment definition). We say $C ◃_{\times} D$ if $C ◃ D$ and $C ◃^{*} D$ . Equivalently, we say $C ◃_{\times} D$ if $C ◃ D$ and $D^{*} ◃ C^{*}$ .

Additive sub-environment. We say $C$ is an additive sub-environment of $D$ , written $C ◃_{+}^{*} D$ , if $D^{*} ◃_{+} C^{*}$ .

Multiplicative sub-environment. We say $C$ is an multiplicative sub-environment of $D$ , written $C ◃_{\times}^{*} D$ , if $D^{*} ◃_{\times} C^{*}$ .

5.2. Ways to Construct Subagents, Sub-Environments, etc.

Committing. Given a set $S \subseteq W$ and a frame $C = (A, E, \cdot)$ over $W$ , we define ${Commit}_{S} (C) = {Commit}^{B} (C)$ and ${Commit}_{∖ S} (C) = {Commit}^{∖ B} (C)$ , where $B \subseteq A$ is given by $B = {a \in A | \forall e \in E, a \cdot e \in S}$ .

Assuming. Given a set $S \subseteq W$ and a frame $C = (A, E, \cdot)$ over $W$ , we define ${Assume}_{S} (C) = {Assume}^{F} (C)$ and ${Assume}_{∖ S} (C) = {Assume}^{∖ F} (C)$ , where $F \subseteq E$ is given by $F = {e \in E | \forall a \in A, a \cdot e \in S} .$

Externalizing. Given a partition $V$ of $W$ , let $v : W \to V$ send each element $w \in W$ to the part that contains it. Given a frame $C = (A, E, \cdot)$ over $W$ , we define ${External}_{V} (C) = {External}^{B} (C)$ and ${External}_{/ V} (C) = {External}^{/ B} (C)$ , where $B = {{a^{'} \in A | \forall e \in E, v (a^{'} \cdot e) = v (a \cdot e)} | a \in A}$ .

Internalizing. Given a partition $V$ of $W$ , let $v : W \to V$ send each element $w \in W$ to the part that contains it. Given a frame $C = (A, E, \cdot)$ over $W$ , we define ${Internal}_{V} (C) = {Internal}^{F} (C)$ and ${Internal}_{/ V} (C) = {Internal}^{/ F} (C)$ , where $F = {{e^{'} \in E | \forall a \in a, v (a \cdot e^{'}) = v (a \cdot e)} | e \in E}$ .

6. Controllables and Observables

Ensurables (categorical definition). $Ensure (C)$ is the set of all $S \subseteq W$ such that there exists a morphism $ϕ : 1_{S} \to C$ .

Preventables (categorical definition). $Prevent (C)$ is the set of all $S \subseteq W$ such that there exists a morphism $ϕ_{1} : 1_{W ∖ S} \to C$ .

Controllables (categorical definition). Let $2_{S}$ denote the Cartesian frame $1_{S} \oplus 1_{W ∖ S}$ . $Ctrl (C)$ is the set of all $S \subseteq W$ such that there exists a morphism $ϕ : 2_{S} \to C$ .

Observables (original categorical definition). $Obs (C)$ is the set of all $S \subseteq W$ such that there exist $C_{0}$ and $C_{1}$ with $Image (C_{0}) \subseteq S$ and $Image (C_{1}) \subseteq W ∖ S$ such that $C ≃ C_{0} & C_{1}$ .

Observables (definition from subsets). We say that a finite partition $V$ of $W$ is observable in a frame $C$ over $W$ if for all parts $S_{i} \in V$ , $S_{i} \in Obs (C)$ . We let ${Obs}^{'} (C)$ denote the set of all finite partitions of $W$ that are observable in $C$ .

Observables (conditional policies definition): We say that a finite partition $V$ of $W$ is observable in a frame $C = (A, E, \cdot)$ over $W$ if for all functions $f : V \to A$ , there exists an element $a_{f} \in A$ such that for all $e \in E$ , $f (v (a_{f} \cdot e)) \cdot e = a_{f} \cdot e$ , where $v : W \to V$ is the function that sends each element of $W$ to its part in $V$ .

Observables (non-constructive additive definition): We say that a finite partition $V = {S_{1}, \dots, S_{n}}$ of $W$ is observable in a frame $C$ over $W$ if there exist frames $C_{1}, \dots C_{n}$ over $W$ , with $C_{i} ◃ ⊥_{S_{i}}$ such that $C ≃ C_{1} & \dots & C_{n}$ .

Observables (constructive additive definition): We say that a finite partition $V = {S_{1}, \dots, S_{n}}$ of $W$ is observable in a frame $C$ over $W$ if $C ≃ {Assume}_{S_{1}} (C) & \dots & {Assume}_{S_{n}} (C)$ .

Powerless outside of a subset: Given a frame $C = (A, E, \cdot)$ over $W$ and a subset $S$ of $W$ , we say that $C$ ’s agent is powerless outside $S$ if for all $e \in E$ and all $a_{0}, a_{1} \in A$ , if $a_{0} \cdot e \notin S$ , then $a_{0} \cdot e = a_{1} \cdot e$ .

Observables (non-constructive multiplicative definition): We say that a finite partition $V = {S_{1}, \dots, S_{n}}$ of $W$ is observable in a frame $C$ over $W$ if $C ≃ C_{1} \otimes \dots \otimes C_{n}$ , where each $C_{i}$ ’s agent is powerless outside $S_{i}$ .

Observables (constructive multiplicative definition): We say that a finite partition $V = {S_{1}, \dots, S_{n}}$ of $W$ is observable in a frame $C$ over $W$ if $C ≃ C_{1} \otimes \dots \otimes C_{n}$ , where $C_{i} = {Assume}_{S_{i}} (C) & 1_{T_{i}}$ , where $T_{i} = (W ∖ S_{i}) \cap Image (C)$ .

Observables (non-constructive internalizing-externalizing definition): We say that a finite partition $V$ of $W$ is observable in a frame $C = (A, E, \cdot)$ over $W$ if either $A = {}$ or $C$ is biextensionally equivalent to something in the image of ${External}_{V} \circ {Internal}_{V}$ .

Observables (constructive internalizing-externalizing definition): We say that a finite partition $V$ of $W$ is observable in a frame $C = (A, E, \cdot)$ over $W$ if either $A = {}$ or $C ≃ {External}_{V} ({Internal}_{V} (C))$ .