Contents

Idea

Twisted K-theory is a twisted cohomology version of K-theory, where the twist is given by a cocycle in degree 3 ordinary cohomology.

Definition

Let $\mathrm{Vectr}$ be the stack of vectorial bundles. (If we just take vector bundles we get a notion of twisted K-theory that only allows twists that are finite order elements in their cohomology group).

There is a canonical morphism

coming from the standard representation of the group $U(1)$.

Let $B_{\otimes }\mathrm{Vectr}$ be the delooping of $\mathrm{Vectr}$ with respect to the tensor product monoidal structure (not the additive structure).

Then we have a fibration sequence

of (infinity,1)-categories (instead of infinity-groupoids).

The entire morphism above deloops

being the standard representation of the 2-group $BU(1)$.

From the general nonsense of twisted cohomology this induces canonically now for every $B^{2}U(1)$-cocycle $c$ (for instance given by a bundle gerbe) a notion of $c$-twisted $\mathrm{Vectr}$-cohomology:

After unwrapping what this means, the result of

• Kiyonori Gomi, Twisted K-theory and finite-dimensional approximation (arXiv)

shows that concordance classes in $H^c(X,\mathrm{Vectr})$ yield twisted K-theory.

References

The basic idea underlying this description of twisted bundles as homotopies from the gerbe to the trivial gerbe but inside a category of “2-vector bundles” was (apparently first?) presented around secton 3.2 of

• Urs Schreiber, Sections of 2-vector bundles (pdf)

A more refined description of this is in section 4.4.3 Twisted vector bundles of

• U. Schreiber, Konrad Waldorf, Connections on non-abelian Gerbes and their Holonomy (arXiv)

The above formulation of the relevant homotopies evolved from the joint work mentioned at twisted cohomology and in particular from interaction with Thomas Nikolaus.