*[HoTT] Orthogonal groups and grassmannians in HoTT@ 2018-09-17 0:20 Ali Caglayan0 siblings, 0 replies; 1+ messages in thread From: Ali Caglayan @ 2018-09-17 0:20 UTC (permalink / raw) To: Homotopy Type Theory [-- Attachment #1.1: Type: text/plain, Size: 1904 bytes --] There is a wonderful paper by Rijke <https://arxiv.org/pdf/1701.07538.pdf> detailing an interesting construction called the "join construction". I would argue that interesting is a vast understatement because it details how to construct images of maps in HoTT. This encapsulates many interesting constructions in HoTT: set quotients, propositional truncation and Rezk completion. Now suppose we have a "oo-group" which can be taken as a pointed conntected type BG with map pt : 1 --> BG. The iterated join construction on pt gives the classical Milnor-construction. In the case where BG = K(Z/2,1) the real projective spaces are obtained (synthetic homotopically mind you) which is expanded on elsewhere with Buchholtz <https://arxiv.org/pdf/1704.05770.pdf>. When BG = K(Z,2) we obtain a definition for the complex projectve spaces. It is hinted at that this construction may be modified to obtiain grassmannians, I would like to know some more details about this. But it would seem to me that BO(n) would need to be known which is kindof circular if one defines it as an infinite grassmannian. Mike has said on Mathoverflow <https://mathoverflow.net/a/289730/54401> that SU(n) would probably be constructed as BSU(n) in HoTT and obiously loop spaced to get an automatic group structure. If we are to define BSU(n) as a complex grassmannian then I don't see how we can do it the Rijke-Buchholtz way. Because then it seems we would have to know BSU(n) anyway. I would like to hear the communities thoughts on this problem and whether there is any proposed solution. - Ali Caglayan -- 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: 2266 bytes --] ^ permalink raw reply [flat|nested] 1+ messages in thread

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