On algebraic K-theory categorical groups (Q543495)

This is about categorical groups related with algebraic \(K\)-theory. 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 \(\mathbb{K}_i R\) of any associated ring \(R\) with 1. The authors also show the existence of a fundamental categorical crossed module associated to any fibre homotopy sequence and then, \(\mathbb{K}_1 R\) and \(\mathbb{K}_2 R\) are characterized, respectively, as the homotopy cokernel and kernel of the fundamental categorical crossed module associated to a fibre homotopy sequence.