[-- Attachment #1: Type: text/plain, Size: 3459 bytes --] This is related to a previous discussion on this mailing list : https://groups.google.com/d/msg/homotopytypetheory/b96TBt0Bn7Y/IiAcnUL2AQAJ, "What is known and/or written about “Frobenius eliminators”?". I noticed recently that the Frobenius eliminator for identity types is related to the notion of strong logical equivalence (the trivial fibrations of the left semi-model structure on the category of CwAs with Sigma and Id introduced in https://arxiv.org/abs/1610.00037,"The homotopy theory of type theories"). The Frobenius, Paulin-Mohring variant of the J eliminator is presented by the following inference rule: G |- A type G |- x : A G, y:A, p:Id(x,y) |- B(y,p) type* G, y:A, p:Id(x,y), b:B(y,p) |- C(y,p,b) type G, b:B(x,refl(x)) |- c : C(x,refl(x),b) type G |- y : A G |- p : Id(x,y) G |- b:B(y,p) -------------------------------------------------------- G |- J(A,x,B,C,c,y,p,b) : C(y,p,b) (where "B(y,p) type*" means that B(y,p) is a finite sequence of types over "G, y:A, p:Id(x,y)", i.e. a context in the contextual slice over "G, y:A, p:Id(x,y)") (In the discussion linked above, Valery Isaev gives a proof that if the type theory includes sigma types, the "simple" Paulin-Mohring eliminator is strong enough to derive the "Frobenius" variant. This is also proven by Paige Randall North in https://arxiv.org/abs/1901.03567, in a more categorical setting.) I consider roughly the same framework as in "The homotopy theory of type theories". Recall that a contextual morphism F : C --> D is a strong logical equivalence if it satisfies term lifting and type lifting conditions: - type lifting: for every context G in C, F_G : Ty_C(G) -> Ty_D(F(G)) is surjective. - term lifting: for every context G in C and type A over G, F_A : Ter_C(G,A) -> Ter_D(F(G),F(A)) is surjective. Let C be a CwA with Id and refl. The J eliminator for some pointed type (A,x) in some context G can be seen as the statement that the contextual morphism F : C[G,y:A,p:Id(x,y)] --> C[G] (where C[G] is a notation for the contextual slice over G), sending (y,p) to (x,refl(x)), satisfies the term lifting condition. Indeed, the input of the term lifting condition is the data of B, C and c, and the output is a term J in the context G,y:A,p:Id(x,y),b:B(y,p) of type C(y,p,b), such that F(J) = c. The type lifting condition can then be derived, assuming that C does not include weird type equalities (for instance, when C is cofibrant/freely generated, or when C is equipped with a hierarchy of universes ensuring that every type belongs to a unique universe). If we only require F to be a weak logical equivalence (a weak equivalence in the left semi-model structure), then we obtain propositional identity types. I think that asking for F to be an isomorphism forces the identity types to become extensional. Has this been studied before ? Can this be extended to the elimination rules of other type formers ? Best regards, Rafaël Bocquet -- 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/a20122bc-289b-4fce-1183-76b66aed395f%40ens.fr. [-- Attachment #2: Type: text/html, Size: 4718 bytes --] <html> <head> <meta http-equiv="content-type" content="text/html; charset=UTF-8"> </head> <body bgcolor="#FFFFFF" text="#000000"> <p>This is related to a previous discussion on this mailing list : <a class="moz-txt-link-freetext" href="https://groups.google.com/d/msg/homotopytypetheory/b96TBt0Bn7Y/IiAcnUL2AQAJ">https://groups.google.com/d/msg/homotopytypetheory/b96TBt0Bn7Y/IiAcnUL2AQAJ</a>, "<span class="F0XO1GC-mb-Z" id="t-t">What is known and/or written about “Frobenius eliminators”?"</span>.<br> </p> <p>I noticed recently that the Frobenius eliminator for identity types is related to the notion of strong logical equivalence (the trivial fibrations of the left semi-model structure on the category of CwAs with Sigma and Id introduced in <a class="moz-txt-link-freetext" href="https://arxiv.org/abs/1610.00037">https://arxiv.org/abs/1610.00037</a>,"The homotopy theory of type theories").</p> <p>The Frobenius, Paulin-Mohring variant of the J eliminator is presented by the following inference rule:<br> G |- A type G |- x : A<br> G, y:A, p:Id(x,y) |- B(y,p) type*<br> G, y:A, p:Id(x,y), b:B(y,p) |- C(y,p,b) type<br> G, b:B(x,refl(x)) |- c : C(x,refl(x),b) type<br> G |- y : A G |- p : Id(x,y) G |- b:B(y,p)<br> --------------------------------------------------------<br> G |- J(A,x,B,C,c,y,p,b) : C(y,p,b)<br> </p> <p>(where "B(y,p) type*" means that B(y,p) is a finite sequence of types over "G, y:A, p:Id(x,y)", i.e. a context in the contextual slice over "G, y:A, p:Id(x,y)")</p> <p>(In the discussion linked above, Valery Isaev gives a proof that if the type theory includes sigma types, the "simple" Paulin-Mohring eliminator is strong enough to derive the "Frobenius" variant. This is also proven by Paige Randall North in <a class="moz-txt-link-freetext" href="https://arxiv.org/abs/1901.03567">https://arxiv.org/abs/1901.03567</a>, in a more categorical setting.)<br> </p> <p>I consider roughly the same framework as in "The homotopy theory of type theories". <br> </p> <p>Recall that a contextual morphism F : C --> D is a strong logical equivalence if it satisfies term lifting and type lifting conditions:<br> - type lifting: for every context G in C, F_G : Ty_C(G) -> Ty_D(F(G)) is surjective.<br> - term lifting: for every context G in C and type A over G, F_A : Ter_C(G,A) -> Ter_D(F(G),F(A)) is surjective.</p> <p>Let C be a CwA with Id and refl. The J eliminator for some pointed type (A,x) in some context G can be seen as the statement that the contextual morphism F : C[G,y:A,p:Id(x,y)] --> C[G] (where C[G] is a notation for the contextual slice over G), sending (y,p) to (x,refl(x)), satisfies the term lifting condition. Indeed, the input of the term lifting condition is the data of B, C and c, and the output is a term J in the context G,y:A,p:Id(x,y),b:B(y,p) of type C(y,p,b), such that F(J) = c. <br> </p> <p>The type lifting condition can then be derived, assuming that C does not include weird type equalities (for instance, when C is cofibrant/freely generated, or when C is equipped with a hierarchy of universes ensuring that every type belongs to a unique universe).<br> </p> <p>If we only require F to be a weak logical equivalence (a weak equivalence in the left semi-model structure), then we obtain propositional identity types. I think that asking for F to be an isomorphism forces the identity types to become extensional.</p> <p>Has this been studied before ? Can this be extended to the elimination rules of other type formers ?<br> </p> <p>Best regards,<br> Rafaël Bocquet<br> </p> </body> </html> <p></p> -- <br /> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br /> To unsubscribe from this group and stop receiving emails from it, send an email to <a href="mailto:HomotopyTypeTheory+unsubscribe@googlegroups.com">HomotopyTypeTheory+unsubscribe@googlegroups.com</a>.<br /> To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/HomotopyTypeTheory/a20122bc-289b-4fce-1183-76b66aed395f%40ens.fr?utm_medium=email&utm_source=footer">https://groups.google.com/d/msgid/HomotopyTypeTheory/a20122bc-289b-4fce-1183-76b66aed395f%40ens.fr</a>.<br />