[-- Attachment #1.1: Type: text/plain, Size: 1859 bytes --] Some time ago (May 2014), there was a discussion on this list about defining the circle as the type of Z-torsors. Dan had formalized in UniMath that this type satisfies the recursion principle for the circle, and Mike posted a quick sketch showing how to extend the proof to give the induction principle: https://groups.google.com/d/msg/homotopytypetheory/hE1eY-v_Kes/bdSoAxC9224J We're now pleased to announce that this indeed works, with a detailed proof developed independently of Mike's sketch by Marc and Ulrik, and formalized in UniMath by Dan: https://github.com/DanGrayson/UniMath/blob/circle/UniMath/SyntheticHomotopyTheory/Circle2.v Definition circle := B ℤ. Definition pt := basepoint circle. Theorem loops_circle : ℤ ≃ Ω circle. Definition loop := loops_circle 1 : Ω circle. Definition CircleInduction (circle : Type) (pt : circle) (loop : pt = pt) := ∏ (X:circle->Type) (x:X pt) (p:PathOver x x loop), ∑ (f:∏ t:circle, X t) (r : f pt = x), apd f loop = r ⟤ p ⟥ !r. Theorem circle_induction : CircleInduction circle pt loop. If the underlying type theory has propositional truncation with an eliminator that computes judgmentally on the point constructor, then the same is true for our circle, i.e., the 'r' above is a reflexivity path. We're working on a detailed write-up of the proof; you'll find a preliminary version of this if you follow a link in the above formalization. Marc Bezem, Ulrik Buchholtz, and Dan Grayson -- 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: 2312 bytes --] <div dir="ltr"><div>Some time ago (May 2014), there was a discussion on this list about defining the circle as the type of Z-torsors. Dan had formalized in UniMath that this type satisfies the recursion principle for the circle, and Mike posted a quick sketch showing how to extend the proof to give the induction principle:</div><div><br></div><div> https://groups.google.com/d/msg/homotopytypetheory/hE1eY-v_Kes/bdSoAxC9224J</div><div><br></div><div>We're now pleased to announce that this indeed works, with a detailed proof developed independently of Mike's sketch by Marc and Ulrik, and formalized in UniMath by Dan:</div><div><br></div><div> https://github.com/DanGrayson/UniMath/blob/circle/UniMath/SyntheticHomotopyTheory/Circle2.v</div><div><br></div><div> Definition circle := B ℤ.</div><div> Definition pt := basepoint circle.</div><div> Theorem loops_circle : ℤ ≃ Ω circle.</div><div> Definition loop := loops_circle 1 : Ω circle.</div><div> Definition CircleInduction (circle : Type) (pt : circle) (loop : pt = pt) :=</div><div> ∏ (X:circle->Type) (x:X pt) (p:PathOver x x loop),</div><div> ∑ (f:∏ t:circle, X t) (r : f pt = x), apd f loop = r ⟤ p ⟥ !r.</div><div> Theorem circle_induction : CircleInduction circle pt loop.</div><div><br></div><div>If the underlying type theory has propositional truncation with an eliminator that computes judgmentally on the point constructor, then the same is true for our circle, i.e., the 'r' above is a reflexivity path.</div><div><br></div><div>We're working on a detailed write-up of the proof; you'll find a preliminary version of this if you follow a link in the above formalization.</div><div><br></div><div>Marc Bezem, Ulrik Buchholtz, and Dan Grayson</div><div><br></div></div> <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 /> For more options, visit <a href="https://groups.google.com/d/optout">https://groups.google.com/d/optout</a>.<br />