*[HoTT] Definitional equality of types as strict bijection@ 2023-02-04 13:56 Madeleine Birchfield0 siblings, 0 replies; only message in thread From: Madeleine Birchfield @ 2023-02-04 13:56 UTC (permalink / raw) To: Homotopy Type Theory [-- Attachment #1.1: Type: text/plain, Size: 3789 bytes --] In this thread, I follow the approach in Egbert Rijke's textbook in having a separate type judgment, rather than the HoTT book's judgment in having an infinite tower of universes and no separate type judgment, and I follow Agda in using the bare equals sign for definitional equality. In structural set theories like William Lawvere's ETCS or Mike Shulman's SEAR it is quite common to have equality only for the sorts of elements, functions, and relations, and not have any primitive equality relation for the sort of sets. This means that the only way to compare sets for "equality" is through a bijection of sets. In addition, Mike Shulman's talks on Higher Observational Type Theory back in May of 2022 includes a section where he talks about the right notion of "definitional univalence" is not using the usual notion of definitional equality of types, but rather strict bijection, where there are functions idtoequiv(a, b):Id_U(a, b) -> Equiv_U(a, b) equivtoid(a, b):Equiv_U(a, b) -> Id_U(a, b) between the identity types Id_U(a, b) and the type of equivalences Equiv_U(a, b) of elements a:U and b:U of a universe U, and definitional equalities of elements equivtoid(a, b)(idtoequiv(a, b)(p)) = p:Id_U(a, b) idtoequiv(a, b)(equivtoid(a, b)(R)) = R:Equiv_U(a, b) Using insights from structural set theory, one could go beyond definitional univalence in Mike Shulman's Higher Observational Type Theory and directly modify the structural rules for definitional equality of types so that it behaves not as an equivalence relation a la first order logic with equality, but rather strict bijection of types a la structural set theory, using the following rules: Gamma |- A = B type --------------------------------- Gamma, x:A |- f(x):B Gamma |- A = B type --------------------------------- Gamma, y:B |- g(y):A Gamma |- A = B type --------------------------------- Gamma, x:A |- g(f(x)) = x:A Gamma |- A = B type --------------------------------- Gamma, y:B |- f(g(y)) = y:B The variable conversion rule in dependent type theory then becomes substitution of g(y) for x, Gamma |- A = A' type Gamma, x:A, Delta(x) |- J(x) ------------------------------------------------------------------------------------ Gamma, y:A', Delta(g(y)) |- J(g(y)) The first congruence rule for substitution is the following rule Gamma |- a = a':A Gamma, x:A, Delta(x) |- B(x) type ------------------------------------------------------------------------------------ Gamma, Delta(a) |- B(a) = B(a') type and the strict bijection rules for definitional equality of types implies that f and g for the definitional equality B(a) = B(a') type are judgmental versions of transport. The second congruence rule for substitution becomes merely a judgmental version of application to paths: Gamma |- a = a':A Gamma, x:A, Delta(x) |- b(x):B(x) type ------------------------------------------------------------------------------------------ Gamma, Delta(a) |- b(a) = g(b(a')):B(a) Definitional univalence then becomes under this collection of structural rules the following rule Gamma |- U type Gamma |- A:U Gamma |- B:U ------------------------------------------------------------------------------------------ Gamma |- Id_U(A, B) = Eq_U(A, B) type from which one could derive Mike Shulman's original 4 rules for definitionally univalent universes. Madeleine Birchfield -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/45bf6ae5-b73b-4df2-a8bc-9aca972797e1n%40googlegroups.com. [-- Attachment #1.2: Type: text/html, Size: 4668 bytes --] ^ permalink raw reply [flat|nested] only message in thread

only message in thread, other threads:[~2023-02-04 13:59 UTC | newest]Thread overview:(only message) (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2023-02-04 13:56 [HoTT] Definitional equality of types as strict bijection Madeleine Birchfield

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