On Algebraic K-theory Categorical Groups

Homotopy categorical groups of any pointed space are defined via the fundamental groupoid of iterated loop spaces. This notion allows, paralleling the group case, to introduce the notion of K-categorical groups KiR of any ring R. We also show the existence of a fundamental categorical crossed module associated to any fibre homotopy sequence and then, K1R and K2R are characterized, respectively, as the homotopy cokernel and kernel of the fundamental categorical crossed module associated to te fibre homotopy sequence F(R) dR // BGL(R) qR // BGL(R)+ . As consequence, the 3th level of the Postnikov tower of the K-theory spectrum of R is classified by this categorical crossed module.