Constructing Twisted K-theory

Question

There is a simple, intuitive "construction" of twisted K-theory if we are allowed to ignore that many things only hold up to homotopy. We know that maps to $K(Z,2)$ give line bundles on a space and that $K(Z,2)$ forms a group corresponding to the tensor product of line bundles. Line bundles also act as endomorphisms of K-theory given by the tensor product. Thus, there is an action of $K(Z,2)$ on $F$ (where $F$ is the classifying space for $K^0$). $K(Z,2)$ principal bundles are classified by maps to $BK(Z,2) \cong K(Z,3)$, ie, elements of $H^3$. Choosing such a map, we get a principal $K(Z,2)$ bundle, $E$, and we can form the associated bundle $E \times_{K(Z,2)} F$. Twisted K-theory is then the homotopy classes of sections of this bundle.

The usual constructions of twisted K-theory that I have seen make the above precise by choosing representatives of the relevant objects so that all the needed relations hold on the nose. My question is whether you can avoid doing that. In other words, can you define all the various notions up to homotopy and obtain a definition of twisted K-theory that way?

Answer 1

The answer is yes if you're working on the level of $\infty$-categories (and I'm pretty sure no if you're working on the level of homotopy categories). In other words, in the $\infty$-world there's no problem talking about a principal bundle for K(Z,2)=BBZ on any space, and they're indeed classified by maps (in the $\infty$-category of spaces) to BBBZ, as are elements of $H^3$. Moreover for any spectrum $E$ there's a well defined group object $GL_1(E)$, and we have a map $BBZ\to GL_1(E)$ in the case of E=K-theory. For any principal $GL_1(E)$ bundle (eg one induced from a BBZ bundle) we have an associated "E line bundle", and its global sections are the twisted K-theory of your space (i.e. a spectrum whose homotopy groups are the usual twisted K-groups).

Or again in short, everything works intuitively as you think without chosing representatives or strictifying if you work in the wonderful world of $\infty$-categories.. for example, the idea of sheaves of spectra (hence twisted cohomology theories) are as simple to work with formally as ordinary sheaves. When it comes to calculating things.. well that's a whole other story.