Are the two frames labeled C0 and C1 in section 3.3 (with an agent that thinks about either red or green and walks or stays home, and an environment that’s safe or has bears) equivalent? (I would guess so, given the section that example was in, but didn’t see it stated explicitly.)
They’re not equivalent. If two frames are ‘homotopy equivalent’ / ‘biextensionally equivalent’ (two names for the same thing, in Cartesian frames), it means that you can change one frame into the other (ignoring the labels of possible agents and environments, i.e., just looking at the possible worlds) by doing some combination of ‘make a copy of a row’, ‘make a copy of a column’, ‘delete a row that’s a copy of another row’, and/or ‘delete a column that’s a copy of another column’.
The entries of C0 and C1 are totally different (Image(C0)={w0,w1,w2,w3,w4,w5,w6,w7}, while Image(C1)={w8,w9,w10,w11}, before we even get into asking how those entries are organized in the matrices), so they can’t be biextensionally equivalent.
There is an important relationship between C0 and C1, which Scott will discuss later in the sequence. But the reason they’re brought up in this post is to make a more high-level point “here’s a reason we want to reify agents and environments less than worlds, which is part of why we’re interested in biextensional equivalence,” not to provide an example of biextensional equivalence.
Are the two frames labeled C0 and C1 in section 3.3 (with an agent that thinks about either red or green and walks or stays home, and an environment that’s safe or has bears) equivalent? (I would guess so, given the section that example was in, but didn’t see it stated explicitly.)
They’re not equivalent. If two frames are ‘homotopy equivalent’ / ‘biextensionally equivalent’ (two names for the same thing, in Cartesian frames), it means that you can change one frame into the other (ignoring the labels of possible agents and environments, i.e., just looking at the possible worlds) by doing some combination of ‘make a copy of a row’, ‘make a copy of a column’, ‘delete a row that’s a copy of another row’, and/or ‘delete a column that’s a copy of another column’.
The entries of C0 and C1 are totally different (Image(C0)={w0,w1,w2,w3,w4,w5,w6,w7}, while Image(C1)={w8,w9,w10,w11}, before we even get into asking how those entries are organized in the matrices), so they can’t be biextensionally equivalent.
There is an important relationship between C0 and C1, which Scott will discuss later in the sequence. But the reason they’re brought up in this post is to make a more high-level point “here’s a reason we want to reify agents and environments less than worlds, which is part of why we’re interested in biextensional equivalence,” not to provide an example of biextensional equivalence.
An example of frames that are biextensionally equivalent to C1:
⎛⎜ ⎜ ⎜⎝w8w9w10w11w8w9w10w11⎞⎟ ⎟ ⎟⎠≃⎛⎜⎝w8w9w10w11w8w9⎞⎟⎠≃⎛⎜⎝w8w9w10w11w10w11⎞⎟⎠≃(w8w9w10w11)
… or any frame that enlarges one of those four frames by adding extra copies of any of the rows and/or columns.
This is helpful. Thanks!
Scott’s post explaining the relationship between C0 and C1 exists as of now: Functors and Coarse Worlds.