The n-Category Café

This is very nice! I’m about halfway through it. I really like the replacement of global “possible worlds” (which I could never really make sense of metaphysically) by a local “domain of variation” or “implicitly understood collection of relevant situations” for every usage of modalities like “necessary” and “possible” in natural language.

One suggestion: I found the notation on p13 to be kind of confusing. You seem to be using “$7(w)$” to denote the “function which sends any argument $w$ to the constant $7$”, which is the opposite of the usual convention by which $f$ denotes a function and $f(x)$ denotes its value at an argument $x$. I had trouble even parsing the more complicated constructions like $\Box_W(8(w)\gt 7(w))(w)$ — is this supposed to mean $\Box_W((8(w)\gt 7(w))(w))$ or $(\Box_W(8(w)\gt 7(w)))(w)$? Do both of those even make sense? What do they mean? In “(the number of planets$(w) \gt 7(w))(a)$” do the two $(w)$s denote simply “making something a function” while the $(a)$ ambiguously denotes evaluating a function at a point $a$?

I suggest a notation for making functions that is different from the notation for evaluating functions. Would your intended audience be able to deal with $\lambda$-notation? If “the number of planets” is implicitly considered a function of world, then the latter example could be written “(the number of planets $\gt_W (\lambda w.7))(a)$” (where $\gt_W$ denotes the pointwise application of $\gt$ to functions), which I find much clearer, and which can be $\beta$-reduced to “(the number of planets)$(a) \gt (\lambda w.7)(a) = 7$”.

I also really like the idea of necessary elements of types as “essential characteristics” on p15:

An element generated in such a way might be said to refer to an essential characteristic of a dog. If I point to the front right leg of a dog and show you another dog, you will probably choose the same leg. It might be holding this paw in the air, so you could have chosen the left rear leg of the second dog who is cocking this now at a lamp post, but it does seem that the same element of the body plan is the most reasonable choice.

It reminds me of some psychological research I’ve encountered on the way even children intuitively generalize in some ways in preference to others. I don’t have any actual citations to point to (maybe some bystander can supply some); my “encounter” consisted of a 5-minute research study that my 4-year-old son participated in at our local science museum.

I don’t remember the details exactly, but I think the researcher had some collection of objects that varied along two axes, maybe size and color, and the kid’s job was to sort them according to whether they made a light bulb (secretly controlled by the experimenter) light up when placed in front of it. For each object the kid was asked first what they thought it would do and why, before placing it in front of the bulb and putting it in the correct pile and then explaining why it behaved that way. I think the actual pattern was that the green objects made the bulb light up and the blue ones didn’t, but supposedly a lot of young children have a deeply embedded idea that “larger things are more effective” which they maintain in the face of evidence that in this case it’s the color rather than the size that matters. Or maybe the other way around?

Anyway, it occurred to me that you could describe what is happening as the children incorrectly guessing the appropriate map with respect to which the modalities should be defined. In your example, when pointing to the right front leg of a dog and asking me to point to the “same” leg of another dog, our experience with the world suggests that it is more useful to consider Fido’s legs as pulled back along $Animal \to Species$ than as pulled back along $Animal \to AnimalWithOneLegInTheAir$. (For instance, one reason for this is that the former map is fiberwise over $Time$ whereas the latter isn’t.) But a hypothetical alien or visitor from another dimension might not be trained to make the same assumption.

Minor comment: I was briefly a bit confused by the introduction of the map $spec : Animal \to Species$ in the middle of page 15, since it seemed like it was introducing something new, but we had already been talking about this map before. Of course soon I realized you were just giving a name at last to this map which had not been previously named. But it might be clearer to name this map when it is first introduced back on p14.