[HoTT] Orthogonal groups and grassmannians in HoTT

[Ali Caglayan <alizter@gmail.com>]

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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

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