# 24

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  for the matrix visualizations below. Let  be an arbitrary Cartesian frame.

## 2. Binary Operations

Sum. For Cartesian frames  and  over  is the Cartesian frame , where  if , and  if .

Product. For Cartesian frames  and  over  is the Cartesian frame , where  if , and  if .

Tensor. Let  and  be Cartesian frames over . The tensor product of  and , written , is given by , where  is the set of morphisms  (i.e., the set of all pairs  such that  for all ), and  is given by .

Par. Let  and  be Cartesian frames over , where .

Lollipop. Given two Cartesian frames over  and , we let  denote the Cartesian frame , where  is given by .

## 3. Frames, Morphisms, and Equivalence Relations

Cartesian frame. A Cartesian frame  over a set  is a triple , where  and  are sets and . If  is a Cartesian frame over , we say , and .

Environment subset. Given a Cartesian frame  over , and a subset  of , let  denote the subset .

Cartesian frame image. .

Chu category.  is the category whose objects are Cartesian frames over , whose morphisms from  to  are pairs of functions , such that  for all  and , and whose composition of morphisms is given by .

Isomorphism. A morphism  is an isomorphism if both  and  are bijective. If there is an isomorphism between  and , we say .

Homotopic. Two morphisms  with the same source and target are called homotopic if  is also a morphism.

Homotopy equivalence / biextensional equivalence.  is homotopy equivalent (or biextensionally equivalent) to , written , if there exists a pair of morphisms  and  such that  is homotopic to the identity on  and  is homotopic to the identity on .

Sub-sum. Let , and let . A sub-sum of C and D is a Cartesian frame of the form , where  and  is  restricted to , such that  and , where  is  restricted to  and  is  restricted to . Let  denote the set of all sub-sums of  and .

Sub-tensor. Let , and let . A sub-tensor of  and  is a Cartesian frame of the form , where  and  is  restricted to , such that  and , where  and  are given by  and . Let  denote the set of all sub-tensors of  and .

## 4. Functors

Functions between worlds. Given a Cartesian frame  over , and a function , let  denote the Cartesian frame over , where .

Dual. Let  be the functor given by , where , and .

Functor (from functions between worlds). Given two sets  and and , and a function , let  denote the functor that sends the object  to the object , where , and sends the morphism  to the morphism with the same underlying functions, .

Functor (from Cartesian frames). Let  be a Cartesian frame over , with . Then  is the functor that sends  to , where , and sends the morphism  to , where .

## 5. Subagents

Subagent (categorical definition). Let  and  be Cartesian frames over . We say that  is a subagent of , written , if for every morphism  there exists a pair of morphisms  and  such that .

Subagent (currying definition). Let  and  be Cartesian frames over . We say that  if there exists a Cartesian frame  over  such that .

Subagent (covering definition). Let  and  be Cartesian frames over . We say that  if for all , there exists an  and a  such that .

Sub-environment. We say  is a sub-environment of , written , if .

Additive subagent (sub-sum definition).  is an additive subagent of , written , if there exists a  and a  with .

Additive subagent (brother definition).  is called a brother to  in  if  for some . We say  if  has a brother in .

Additive subagent (committing definition). Given Cartesian frames  and  over , we say  if there exist three sets , and , with , and a function  such that  and , where  and  are given by  and .

Additive subagent (currying definition). We say  if there exists a Cartesian frame  over  with , such that .

Additive subagent (categorical definition). We say  if there exists a single morphism  such that for every morphism  there exists a morphism  such that  is homotopic to  .

Multiplicative subagent (sub-tensor definition).  is a multiplicative subagent of , written , if there exists a  and  with .

Multiplicative subagent (sister definition).   is called a sister to  in  if  for some . We say  if  has a sister in .

Multiplicative subagent (externalizing definition). Given Cartesian frames  and  over , we say  if there exist three sets , and , and a function  such that  and , where  and  are given by  and .

Multiplicative subagent (currying definition). We say  if there exists a Cartesian frame  over  with , such that .

Multiplicative subagent (categorical definition). We say  if for every morphism , there exist morphisms  and  such that , and for every morphism , there exist morphisms  and  such that .

Multiplicative subagent (sub-environment definition). We say  if  and . Equivalently, we say  if  and .

Additive sub-environment. We say  is an additive sub-environment of , written , if .

Multiplicative sub-environment. We say  is an multiplicative sub-environment of , written , if .

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

Committing. Given a set  and a frame  over , we define  and , where  is given by .

Assuming. Given a set  and a frame  over , we define  and , where  is given by

Externalizing. Given a partition  of , let  send each element  to the part that contains it. Given a frame  over , we define  and , where .

Internalizing. Given a partition  of , let  send each element  to the part that contains it. Given a frame  over , we define  and , where .

## 6. Controllables and Observables

Ensurables (categorical definition).  is the set of all  such that there exists a morphism .

Preventables (categorical definition).  is the set of all  such that there exists a morphism .

Controllables (categorical definition). Let  denote the Cartesian frame  is the set of all  such that there exists a morphism .

Observables (original categorical definition).  is the set of all  such that there exist  and  with  and  such that .

Observables (definition from subsets). We say that a finite partition  of  is observable in a frame  over  if for all parts . We let  denote the set of all finite partitions of  that are observable in .

Observables (conditional policies definition): We say that a finite partition  of  is observable in a frame  over  if for all functions , there exists an element  such that for all , where   is the function that sends each element of  to its part in .

Observables (non-constructive additive definition): We say that a finite partition  of  is observable in a frame  over  if there exist frames  over , with  such that .

Observables (constructive additive definition): We say that a finite partition  of  is observable in a frame  over  if .

Powerless outside of a subset: Given a frame  over  and a subset  of , we say that 's agent is powerless outside  if for all  and all , if , then .

Observables (non-constructive multiplicative definition): We say that a finite partition  of  is observable in a frame  over  if , where each 's agent is powerless outside .

Observables (constructive multiplicative definition):  We say that a finite partition  of  is observable in a frame  over  if , where , where .

Observables (non-constructive internalizing-externalizing definition): We say that a finite partition  of  is observable in a frame  over  if either  or  is biextensionally equivalent to something in the image of .

Observables (constructive internalizing-externalizing definition): We say that a finite partition  of  is observable in a frame  over  if either  or .

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