makes sense; so I guess it isn’t pointing to any particular value; I was having trouble thinking of an interesting constant anywhere in that range

I thought, what with it being a pseudojoke so deeply buried in mathematics and jargon that people could only plausibly get it accidentally, that its absurd affect was kind of the one I like to try and do every so often here, and therefore it seemed like a good avatar.

Or something.

twocubes:

wait, so, then, there must be a similar generalization of measurable space and measurable function.

Like, idk, it’s a ~measurable morphism~ if its left adjoint preserves all limits and colimts?

And the categories are complete and cocomplete and… something or other…

Turns out, for morphisms, this is apparently called “being an essential geometric morphism”. Who knew.

Still unclear about the spaces though…

Maybe I would feel like the logical side of the picture didn't show me how you could do lots of cool stuff with toposes if it weren't for Isham et al.'s topos-theoretic work on quantum mechanics.

Hm. Well, still, these things seem to imply that there is quite probably at least Another Cool Thing to be seen going through a geometric approach. I mean, these ideas did come out of geometry, right? And that’s still weird, and I think demystifying that is probably a worthwhile project.

wait, so, then, there must be a similar generalization of measurable space and measurable function.

Like, idk, it’s a ~measurable morphism~ if its left adjoint preserves all limits and colimts?

And the categories are complete and cocomplete and… something or other…

thesummerofmark replied to your post “aradial-symmetry replied to your post:differentialprincess replied to…”

idk I learned topos theory from the elementary/logical point of view - in terms of “generalized universes of sets” rather than “generalized spaces”

eh, well, this is a warning I’ve heard on this page from Baez:

many toposophers complain that it’s not substantial enough - it shows how topoi illuminate concepts from logic, but it doesn’t show you can do lots of cool stuff with topoi.

also in this entire paper, and also from the assistant I asked about this project (who I only managed to convince to sanction this project by saying I wanted a more geometric approach…)

so I figured there had to be something to it :V

are you talking about elementary topoi (or whatever the things that come up in logic are) rather than Grothendieck topoi (which are categories of sheaves, and iirc are a special case of the first thing) I guess?

I don’t know what I’m talking about, tbh. I was hoping to see Grothendieck topoi before seeing elementary topoi because my understanding is that elementary topoi are Boring if you haven’t seen Grothendieck topoi first (i mean the whole thing about them is HOW DID THIS COME OUT OF GEOMETRY so if you just see them immediately as generalized set theories you never get to see the good stuff/develop the actually interesting intuition) but yeah, I guess I mean elementary topoi?

so how do you have a function defined on objects you haven’t defined

well, I know they’re categories, I guess? which I would guess would have to be finitely complete and cocomplete, but idk. They have other things about them? A subobject classifier exists?

but, that and the fact that I’d heard they were “generalized spaces” was enough to make me guess correctly…

oh OH

I just understood/guessed correctly

A continuous function f:X → Y between two topological spaces is such that the inverse image function f^{-1}: O(Y) → O(X) preserves finite intersections and arbitrary unions.

A geometric morphism f:X → Y between two toposes is such that its left adjoint f*:Y → X preserves finite limits and arbitrary colimits.

Which is a generalization.

uh… I should probably get to the point where I actually get to see the definition of a topos. But that’s like two more volumes after this one… why have I been working so little, I’m a terrible student…