Homotopy bundle gerbes

The goal of this paper is to define twisting cocycles for higher twisted К-theory. For this we generalize the approach to the twisted К-theory based on the notion of a bundle gerbe. In the present paper we define the notion of a homotopy bundle gerbe related to a homotopy transition cocycle with values in the monoid of endomorphisms of the direct limit of matrix algebras. On the set of such objects over a fixed base Х we define a stable equivalence relation such that its equivalence classes are in one-to-one correspondence with homotopy classes of maps to the localization of the delooping of BU.