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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.

##### MSC:
 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
##### Keywords:
groupoid action; crossed module; automorphism; section; homotopy
GAP; Gpd; XMod
Full Text:
##### References:
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