topoi

Main Unsolved Problems Some of the main unsolved problems in topos theory involve: Classifying all possible topoi: Understanding the full landscape of possible topoi and their interrelationships remains a complex challenge.

Homotopy theory in the context of topoi: Developing a deeper understanding of homotopy theory within the framework of higher topos theory.

And a huge number of structures in math are either Cartesian closed categories, or at least topoi, which are closely related.

And you can take sets as something built on top of categories (specifically, topoi), so category theory is more foundational than set theory.

Imagine trying trying to describe a monoidal category or topoi in the OOP style, it's like trying to describe what a Fourier transform is to someone who insists on using roman numerals.

It progresses smoothly all the way from first principles up to (if you buy all three volumes) sheaves and topoi, but it is kind of expensive.

It also provided the springboard for Grothendieck's developments of topoi.

I think topoi are cool, but I don't think it's a complexity worth thinking about for everyone else who isn't concerned about these things.

Categorical logic is larger than elementary topoi.

I'm not familiar with topos theory, so can someone suggest a setting in which the richer mathematical structure afforded by topoi pays off, since they are presumably more expensive than traditional Euclidean models?

Invert our notions of Aleph/Beth/Betti numbers as some sort of triadic Grothendieck topoi that encode our human brain's sensory instruments that nucleate discrete samples of continuum of reality (ontology) Then every modal logic becomes some mapping of 2^(N) to some set of statements.

The continuous symmetry I think makes it easy to think about diffeomorphisms from different topoi by extracting homeomorphisms from gauge invariant symmetries (in the discrete case it's a lattice, but in the continuous we'd have to formalize some notion of liquid or fluid bases?

Of course that is a benefit in the "meta" level, but it is not like ZFC is a better foundation than topoi/types/etc..

So, it seems that the steps you describe, until and including "Find other naturally occuring examples of topoi" can all be done in first-order logic set theory, is that right?

Linked article reminded me of a passage in "Topoi - The Categorial Analysis of Logic" by Robert Goldblatt: "It would be somewhat misleading to infer [...] that foundational systems act primarily as a basis out of which mathematics is actually created.

Topoi (plural of topos): Categories that satisfy certain axioms making them similar to the category of sets.

Have you read the more modern expositions of these independence proofs based on various Topoi?

Topoi, a concept in category theory, also bridge the two fields.