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    • CommentRowNumber1.
    • CommentAuthortomr
    • CommentTimeMay 14th 2023
    HoTT - homotopy type theory - can represent all the mathematics and HoTT has categorical interpretation (category of contexts, classifying category). E.g. some mathematical theory is collection of HoTT types (or HoTT type itself) and hence some mathematical theory is subcategory of the classifying category. Are there efforts to research all the mathematics as the hierarchy and classification of the subcategories of the classifying category? E.g. reverse mathematics is trying to reconstruct sets of axioms and necessary theorems for any theorem. Maybe combination of reverse mathematics with category theory or doing the reverse mathematics in category theory be the way to represent all the mathematics with categories?

    I am reading and thinking about this.

    Of course, any individual effort in the mathematics that uses categorical tools and any research in category theory, is one step of doing this program already. But I just wanted to know about references that consider this as large scale, intentional and systematic effort?
    • CommentRowNumber2.
    • CommentAuthortomr
    • CommentTimeMay 14th 2023
    I am aware of older discussion and also, that HoTT and second-order-arithmetic are based on different logics, but GPT4 suggested to look on "system of "homotopy type theory with classical logic and choice", studied by Nicolai Kraus, Martín Escardó" (I am still checking this, because it can be hallucination of GPT4), and it is known, that SOA can be interpreted in topos, or at least in the category of sets and that is why some connection with classifying category can still be established, maybe subcategory-category, maybe not.
    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 14th 2023

    A topos basically gives you all of higher-order arithmetic, at minimum. You’d need some kind of arithmetic pretopos with maybe a power object for the parametrised nno, but nothing else, to do SOA