Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthortomr
   • CommentTimeMay 14th 2023
   • PermaLink
   Author: tomr
   Format: TextHoTT - 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 https://www.cambridge.org/core/books/abs/new-spaces-in-mathematics/homotopy-type-theory-the-logic-of-space/D13957048F50112B531698F6FB6269EB 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?

   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 https://www.cambridge.org/core/books/abs/new-spaces-in-mathematics/homotopy-type-theory-the-logic-of-space/D13957048F50112B531698F6FB6269EB 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?
2. • CommentRowNumber2.
   • CommentAuthortomr
   • CommentTimeMay 14th 2023
   • PermaLink
   Author: tomr
   Format: TextI am aware of older discussion https://nforum.ncatlab.org/discussion/7209/reverse-mathematics/ 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.

   I am aware of older discussion https://nforum.ncatlab.org/discussion/7209/reverse-mathematics/ 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.
3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 14th 2023
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexA 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

   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