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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
Software:
GAP; Gpd; XMod
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References:
[1] Alp, M.: GAP, crossed modules, cat1-groups: applications of computational group theory. PhD thesis, University of Wales, Bangor (1997)
[2] Alp, M., Wensley, C.D.: Enumeration of cat1-groups of low order. Internat. J. Algebra Comput. 10, 407–424 (2000) · Zbl 1006.18002
[3] Alp, M., Wensley, C.D.: XMod: crossed modules and cat1-groups in GAP, version 2.12. http://www.maths.bangor.ac.uk/chda/xmod/ (2008)
[4] Baez, J.C., Lauda, A.D.: Higher-dimensional algebra V: 2-groups. Theory Appl. Categ. 12, 423–491 (2004) · Zbl 1056.18002
[5] Brown, R.: Topology and Groupoids. Booksurge PLC, North Charleston (2006) · Zbl 1093.55001
[6] Brown, R., Gilbert, N.D.: Algebraic models of 3-types and automorphism structures for crossed modules. Proc. London Math. Soc. (3) 59, 51–73 (1989) · Zbl 0645.18007
[7] Brown, R., Higgins, P.J.: On the algebra of cubes. J. Pure Appl. Algebra 21, 233–260 (1981) · Zbl 0468.55007
[8] Brown, R., Higgins, P.J.: Tensor product and homotopies for {\(\omega\)}-groupoids and crossed complexes. J. Pure Appl. Algebra 47, 1–33 (1987) · Zbl 0621.55009
[9] Brown, R., Higgins, P.J., Sivera, R.: Nonabelian algebraic topology. A draft of part 1 is available from: http://www.bangor.ac.uk/ mas010/nonab-a-t.html (2008)
[10] Brown, R., \.Içen, I.: Homotopies and automorphisms of crossed modules of groupoids. Appl. Categ. Structures 11, 185–206 (2003) · Zbl 1029.20025
[11] Brown, R., Moore, E.J., Porter, T., Wensley, C.D.: Crossed complexes, and free crossed resolutions. for amalgamated sums and HNN-extensions of groups. Georgian Math. J. 9, 623–644 (2002) · Zbl 1030.55013
[12] Brown, R., Morris, I., Shrimpton, J., Wensley, C.D.: Graphs of morphisms of graphs. Electron. J. Combin. 15(A1), 1–28 (2008) · Zbl 1187.18002
[13] Ellis, G., Steiner, R.: Higher dimensional crossed modules and the homotopy groups of (n+1)-ads. J. Pure Appl. Algebra 46, 117–136 (1987) · Zbl 0622.55010
[14] The GAP Group: GAP–groups, algorithms, programming, version 4.4. http://www.gap-system.org (2004)
[15] Gilbert, N.D.: Derivations, automorphisms and crossed modules. Comm. Algebra 18, 2703–2734 (1990) · Zbl 0704.20045
[16] Higgins, P.J.: Categories and Groupoids. Reprints in Theory and Applications of Categories, vol. 7 (2005). Originally published by Van Nostrand Reinhold, New York (1971) · Zbl 0226.20054
[17] Kamps, K.H., Porter, T.: 2-groupoid enrichments in homotopy theory and algebra. K-Theory 25, 373–409 (2002) · Zbl 1009.18007
[18] Kamps, K.H., Porter, T.: Abstract Homotopy and Simple Homotopy Theory. World Scientific, Singapore (1997) · Zbl 0890.55014
[19] Lue, A.S.-T.: Semi-complete crossed modules and holomorphs of groups. Bull. London Math. Soc. 11, 8–16 (1979) · Zbl 0416.20030
[20] Moore, E.J., Wensley, C.D.: Gpd: groupoids, graphs of groups, and graphs of groupoids, version 1.05 (2008). http://www.maths.bangor.ac.uk/chda/gpd/
[21] Noohi, B.: On weak maps between 2-groups. Available as arXiv:math/0506313v3 (2008). Accessed 13 Jul 2008
[22] Norrie, K.J.: Actions and automorphisms of crossed modules. Bull. Soc. Math. France 118, 129–146 (1990) · Zbl 0719.20018
[23] Roberts, D.M., Schreiber, U.: The inner automorphism 3-group of a strict 2-group. Available as arXiv:0708.1741v2 (2008). Accessed 3 Mar 2008 · Zbl 1186.18004
[24] Wensley, C.D.: Notes on higher-dimensional groups and related topics. http://www.maths.bangor.ac.uk/chda/notes.pdf (2008). Accessed 23 Dec 2008
[25] Whitehead, J.H.C.: On operators in relative homotopy groups. Ann. Math. 49, 610–640 (1948) · Zbl 0041.10102
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