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Automorphisms and homotopies of groupoids and crossed modules. (English) Zbl 1225.18008
Authors’ abstract: This paper is concerned with the algebraic structure of groupoids and crossed modules of groupoids. We describe the group structure of the automorphism group of a finite connected groupoid \({\mathcal C}\) as a quotient of a semidirect product. We pay particular attention to the conjugation automorphisms of \({\mathcal C}\), and use these to define a new notion of groupoid action. We then show that the automorphism group of a crossed module of groupoids \({\mathcal C}\), in the case when the range groupoid is connected and the source group totally disconnected, may be determined from that of the crossed module of groups \({\mathcal C}_u\) formed by restricting to a single object \(u\). Finally, we show that the group of homotopies of \({\mathcal C}\) may be determined once the group of regular derivations of \({\mathcal C}_u\) is known.

18D35 Structured objects in a category (MSC2010)
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
55U99 Applied homological algebra and category theory in algebraic topology
GAP; Gpd; XMod
Full Text: DOI
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