*[HoTT] Homotopy type of simply connected spaces.@ 2019-01-10 20:36 Brian Sanderson2019-01-10 21:11 ` Michael Shulman 0 siblings, 1 reply; 4+ messages in thread From: Brian Sanderson @ 2019-01-10 20:36 UTC (permalink / raw) To: Homotopy Type Theory [-- Attachment #1.1: Type: text/plain, Size: 690 bytes --] The type of a simply connected space would seem to make it just a set as any two paths with the same endpoints would be homotopic. I see that there would not be a continuous function from the space of pairs of paths to homotopies between them. What would the type of a simply connected space look like? Can I say in type theory any two equalities are equal without having a function? -- 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. For more options, visit https://groups.google.com/d/optout. [-- Attachment #1.2: Type: text/html, Size: 890 bytes --] ^ permalink raw reply [flat|nested] 4+ messages in thread

*Re: [HoTT] Homotopy type of simply connected spaces.2019-01-10 20:36 [HoTT] Homotopy type of simply connected spaces Brian Sanderson@ 2019-01-10 21:11 ` Michael Shulman2019-01-11 11:49 ` Brian Sanderson 0 siblings, 1 reply; 4+ messages in thread From: Michael Shulman @ 2019-01-10 21:11 UTC (permalink / raw) To: Brian Sanderson;+Cc:Homotopy Type Theory Yes, you have to truncate the equality. See section 7.5 of the HoTT Book, and also Exercise 7.6. On Thu, Jan 10, 2019 at 12:36 PM Brian Sanderson <brianjsanderson@gmail.com> wrote: > > The type of a simply connected space would seem to make it just a set as any two paths with the same endpoints would be homotopic. I see that there would not be a continuous function from the space of pairs of paths to homotopies between them. What would the type of a simply connected space look like? Can I say in type theory any two equalities are equal without having a function? > > -- > 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. > For more options, visit https://groups.google.com/d/optout. -- 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. For more options, visit https://groups.google.com/d/optout. ^ permalink raw reply [flat|nested] 4+ messages in thread

*Re: [HoTT] Homotopy type of simply connected spaces.2019-01-10 21:11 ` Michael Shulman@ 2019-01-11 11:49 ` Brian Sanderson2019-01-11 12:01 ` Cory Knapp 0 siblings, 1 reply; 4+ messages in thread From: Brian Sanderson @ 2019-01-11 11:49 UTC (permalink / raw) To: Homotopy Type Theory [-- Attachment #1.1: Type: text/plain, Size: 1508 bytes --] Thanks for the references. So am I allowed to say a type is simply connected if any two paths are equal, or is that a meta statement which has no meaning within type theory. On Thursday, 10 January 2019 21:12:13 UTC, Michael Shulman wrote: > > Yes, you have to truncate the equality. See section 7.5 of the HoTT > Book, and also Exercise 7.6. > > On Thu, Jan 10, 2019 at 12:36 PM Brian Sanderson > <brianjs...@gmail.com <javascript:>> wrote: > > > > The type of a simply connected space would seem to make it just a set as > any two paths with the same endpoints would be homotopic. I see that there > would not be a continuous function from the space of pairs of paths to > homotopies between them. What would the type of a simply connected space > look like? Can I say in type theory any two equalities are equal without > having a function? > > > > -- > > 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 <javascript:>. > > > For more options, visit https://groups.google.com/d/optout. > -- 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. For more options, visit https://groups.google.com/d/optout. [-- Attachment #1.2: Type: text/html, Size: 2507 bytes --] ^ permalink raw reply [flat|nested] 4+ messages in thread

*2019-01-11 11:49 ` Brian SandersonRe: [HoTT] Homotopy type of simply connected spaces.@ 2019-01-11 12:01 ` Cory Knapp0 siblings, 0 replies; 4+ messages in thread From: Cory Knapp @ 2019-01-11 12:01 UTC (permalink / raw) To: Brian Sanderson;+Cc:Homotopy Type Theory [-- Attachment #1: Type: text/plain, Size: 2007 bytes --] Using the language of the hott book, a type is simply connected if there *merely exists* a homotopy between any two paths. On Fri, Jan 11, 2019, 11:49 Brian Sanderson <brianjsanderson@gmail.com> wrote: > Thanks for the references. So am I allowed to say a type is simply > connected if any two paths are equal, or is that a meta statement which has > no meaning within type theory. > > > On Thursday, 10 January 2019 21:12:13 UTC, Michael Shulman wrote: >> >> Yes, you have to truncate the equality. See section 7.5 of the HoTT >> Book, and also Exercise 7.6. >> >> On Thu, Jan 10, 2019 at 12:36 PM Brian Sanderson >> <brianjs...@gmail.com> wrote: >> > >> > The type of a simply connected space would seem to make it just a set >> as any two paths with the same endpoints would be homotopic. I see that >> there would not be a continuous function from the space of pairs of paths >> to homotopies between them. What would the type of a simply connected space >> look like? Can I say in type theory any two equalities are equal without >> having a function? >> > >> > -- >> > 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. >> > For more options, visit https://groups.google.com/d/optout. >> > -- > 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. > For more options, visit https://groups.google.com/d/optout. > -- 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. For more options, visit https://groups.google.com/d/optout. [-- Attachment #2: Type: text/html, Size: 2908 bytes --] ^ permalink raw reply [flat|nested] 4+ messages in thread

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