Wait, "horizons clitch'"? What the heck is that apostrophe doing there? Was that intentional?
And oh yeah, the bottomless pit greentext. That was pretty impressive.
Besides isomorphisms and equality of objects, do category theorists use other notions of "equality"?
I'm confused. In any FOL, you have a bunch of "logical axioms" which come built-in to the language, and axioms for whatever theories you want to investigate in said language. You need these or else you've got no way prove basically anything in the language, since your deduction rules are: state an axiom from your logical axioms, from your assumed theory's axioms, or Modus Ponens. And the logical axioms include a number of axioms schemas, such as the ones for equality that I describe, no?
Actually you have just described the same thing twice. There are actually fewer distance-preserving maps than there are continuous ones, and restricting to distance-preserving maps removes all the isomorphisms between the sphere and the cube.
That is a very good point. Hmm. So it seems just plain false that you can break equivalence between two objects by enriching the number of maps between them?
Sorry, I used the word "definition" sloppily there. I don't think we disagree with each other.
I meant something closer to "how equality is formalized in first order logic". That's what the bit about the axiom schemas was referencing: it's how we bake in all the properties we require of the special binary predicate "=". There's a big, infinite core of axiom schemas specifying how "=" works that's retained across FOLs, even as you add/remove character, relation and function symbols to the language.
Yeah, that sure does seem related. Thinking about it a bit more, it feels like equality refers to a whole grab-bag of different concepts. What separates them, what unites them and when they are useful are still fuzzy to me.
Thank you, that is clearly correct and I'm not sure why I made that error. Perhaps because equivalence seems more interesting in category theory than in set theory? Which is interesting. Why is equivalence more central in category theory than set theory?
No worries! For more recommendations like those two, I'd suggest having a look at "The Fast Track" on Sheafification. Of the books I've read from that list, all were fantastic. Note that site emphasises mathematics relevant for physics, and vice versa, so it might not be everyone's cup of tea. But given your interests, I think you'll find it useful.
- Started reading [Procesi] to learn invariant theory and representation theory because it came up quite often as my bottleneck in my recent work (eg). Also interpretability, apparently. So far I just read pg 1-9, reviewing the very basics of group action (e.g., orbit stabilizer theorem). Lie groups aren't coming up until pg ~50 so until then I should catch up on the relevant Lie group prerequisites through [Lee] or [Bredon].
Woit's "Quantum Theory, Groups and Representations" is fantastic for this IMO. It gives physical motivation for representation theory, connects it to invariants and, of course, works through the physically important lie-groups. The intuitions you build here should generalize. Plus, it's well written.
Also, if you are ever in the market for differential topology, algebraic topology, and algebraic geometry, then I'd recommend Ronald Brown's "Topology and Groupoids." It presents the basic material of topology in a way that generalizes better to the fields above, along with some powerful geometric tools for calculations.
Both author's provide free pdfs of their books.
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