Abstract
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 \(\mathbb{K}_iR\) of any ring R. We also show the existence of a fundamental categorical crossed module associated to any fibre homotopy sequence and then, \(\mathbb{K}_1R\) and \(\mathbb{K}_2R\) are characterized, respectively, as the homotopy cokernel and kernel of the fundamental categorical crossed module associated to the fibre homotopy sequence \(FR\xrightarrow{{d_{R} }}BGLR\xrightarrow{{q_{R} }}BGLR^{ + } \) As consequence, the 3th level of the Postnikov tower of the K-theory spectrum of R is classified by this categorical crossed module.
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We thank to the referee the useful observations that greatly improved our exposition.
Partially supported by DGI of Spain and FEDER (Project: MTM2007-65431); Consejería de Innovación de J. de Andalucía (P06-FQM-1889); MEC de España, ‘Ingenio Mathematica(i-Math)’ No.CSD2006-00032 (consolider-Ingenio 2010).
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Garzón, A.R., del Río, A. On Algebraic K-theory Categorical Groups. Appl Categor Struct 19, 633–649 (2011). https://doi.org/10.1007/s10485-009-9195-2
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DOI: https://doi.org/10.1007/s10485-009-9195-2