There is a wonderful paper by Rijke 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 .
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 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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