Cartesian Frames Definitions

by Rob Bensinger6 min read8th Nov 2020No comments

24

Embedded AgencyAIRationality
Frontpage

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.

 visualizationdefinitionnotes
 , where  is empty,  is any singleton set, and  is trivial.. Initial. Identity of sum ().
, where  is any singleton set,  is empty, and  is trivial.. Terminal. . Identity of product ().
, where  and  for all  is the frame .Identity of tensor ().
, where  and  for all  is the frame .. Identity of par ().
, with empty agent, environment, and evaluation function. 

 

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 .

 

5.1. Additive and Multiplicative Subagents

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 .

24

New Comment