@DenisNardin Yeah I would like to know the deloopings, or at least know a method to figure them out. I am not sure if I understand the relationship between postnikov truncations and the omega spectrum for ku. One thing that also confuses is exactly how the double loops of BU is BU, when classical Bott periodicity says double loops on BU is BU*Z

@Deon why do you say "the double loops of BU is BU"? (just in case you weren't here: check the log, just a few days ago I was asking about ku, too)

@Deon So, it is not true that ΩBU = BU. In fact ΩBU=BU×Z. To expand a little, we are looking for an (k-1)-connected pointed space X such that Ω^k X=BU×Z. Let us assume k is even. We know Ω^k U = BU×Z by Bott periodicity. The only problem is that U might not be (k-1)-connected. But that's not a big deal, the space we are looking for is the (k-1)-connected cover of U (since taking (k-1)-connected covers does not change Ω^k). Similarly if k is odd you take the (k-1)-connected cover of BU×Z.

I said Postnikov tower, but really the thing you want to understand is the Whitehead tower of U, which is the "dual" to the Postnikov tower (so knowing one is more or less the same thing as knowing the other)

And when I wrote ΩBU=BU×Z of course I meant Ω²BU = BU×Z. Beware of typos!

Let M be a stable model category enriched in Ch(Ab) and Let M_{\infty} the underline \i- category. Clearly if X is cofibrant and Y is fibrant then the complex Hom_M(X,Y) viewed as a spectrum is the spectral \i-categorical Hom_{M_{\infty}}(X,Y). Is there any reference ?

Since the Hilbert space H is infinite-dimensional, the spaces of linear isometric embeddings of direct sums of H into itself are all contractible, carry free actions of the symmetric groups, and naturally act on products of the space of Fredholms with itself. This gives an explicit $E_\infty$-operad acting on a space homotopy equivalent to $BU \times \mathbb{Z}$ by the Atiyah-Janich theorem

@AaronRoyer Right! I think you can even get explicit deloopings by choosing Clifford-linear Fredholm operators. I always forget about those wacky things...

Yup, there's a paper by Atiyah and Singer about doing exactly that. Dan Freed also has some nice course notes written about it

ma.utexas.edu/users/dafr/M392C/index.html