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>＜Lawvere流の非大卒でも楽しく学べる圏論＞

https://ncatlab.org/nlab/show/topos
topos
Contents
1. Idea
‘What a topos is like:’
2. Definitions
Elementary toposes
Grothendieck/sheaf toposes
W-toposes
Toposes over a base
Morphisms of toposes
3. Properties
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1. Idea
There are various different perspectives on the notion of topos. One is that a topos is a category that looks like a category of spaces that sit by local homeomorphisms over a given base space: all spaces that are locally modeled on a given base space.
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https://mathoverflow.net/questions/9269/category-of-categories-as-a-foundation-of-mathematics
mathoverflow
Category of categories as a foundation of mathematics
asked Dec 18, 2009 Marc Nieper-Wißkirchen
In
Lawvere, F. W., 1966, “The Category of Categories as a Foundation for Mathematics”, Proceedings of the Conference on Categorical Algebra, La Jolla, New York: Springer-Verlag, 1–21.

Lawvere proposed an elementary theory of the category of categories which can serve as a foundation for mathematics.

So far I have heard from several sources that there are some flaws with this theory so that it does not completely work as proposed.

So my question is whether there is currently any (accepted) elementary theory of the category of categories that is rich enough so that one can formulate, say, the following things in the theory:

The category of sets.
Basic notions of category theory (functor categories, adjoints, Kan extensions, etc.).
Other important categories (like the category of rings or the category of schemes).
The elementary theory I am looking for should allow me to identify what should be called a category of commutative rings (at best I would like to see this category defined by a universal 2-categorical property) or how to work with this category. I am not interested in defining groups, rings, etc. as special categories as this seems to be better done in an elementary theory of sets.

P.S.: The same question has an analogue one level higher. Assume that we have constructed an object in the category of categories (=: CAT) which can serve as a, say, category C of spaces. Classically, we can associate to each space X in C the sheaf topos over it. In the picture I have in mind, one should ask whether there is a similar elementary theory of the category of 2-categories (=: 2-CAT). Then one should be able to lift the object C from CAT to 2-CAT (as one is able to form the discrete category from a set), define an object T in 2-CAT that serves as the 2-category of toposes, and a functor C -> T in 2-CAT.

Answers
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