Abstract
The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat1-group or crossed module. An advantage of our construction is that the double groupoid can give an algebraic model of a foliated bundle.
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Al-Agl, F. A., Brown, R. and Steiner, R.: Multiple categories: The equivalence between a globular and cubical approach, Adv. Math. 170 (2002), 71-118.
Borceux, F. and Janelidze, G.: Galois Theories, Cambridge Studies in Advanced Mathematics 72, Cambridge University Press, 2001.
Brown, R.: Groupoids and crossed objects in algebraic topology, Homotopy, Homology Appl. 1 (1999), 1-78.
Brown, R. and Higgins, P. J.: On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978), 193-211.
Brown, R. and Higgins, P. J.: On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), 233-260.
Brown, R. and Higgins, P. J.: Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra 22 (1981), 11-41.
Brown, R. and Higgins, P. J.: The equivalence of ∞-groupoids and crossed complexes, Cahiers Topologie Géom. Différentielle Catég. 22 (1981), 371-386.
Brown, R. and Janelidze, G.: Van Kampen theorems for categories of covering morphisms in lextensive categories, J. Pure Appl. Algebra 119 (1997), 255-263.
Brown, R. and Janelidze, G.: Galois theory of second order covering maps of simplicial sets, J. Pure Appl. Algebra 135 (1999), 83-91.
Brown, R. and Loday, J.-L.: Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311-334.
Brown, R. and Mackenzie, K. C. H.: Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra 80 (1992), 237-271.
Čech, E.: Höherdimensionale Homotopiegruppen, in Verhandlungen des Internationalen Mathematiker-Kongresses Zürich, Band 2, 1932, p. 203.
Dawson, R. and Paré, R.: General associativity and general composition for double categories, Cahiers Topologie Géom. Différentielle Catég. 34 (1993), 57-79.
Ehresmann, C.: Catégories structurées III: Quintettes et applications covariantes, Cahiers Topologie Géom. Différentielle Catég. 5 (1963), 1-22.
Gabriel, P. and Zisman, M.: Calculus of Fractions and Homotopy Theory, Springer, Berlin, 1967.
Janelidze, G.: Precategories and Galois theory, in Lecture Notes in Mathematics 1488, Springer, 1991, pp. 157-173.
Janelidze, G.: Pure Galois theory in categories, J. Algebra 132 (1990), 270-286.
Kamps, K. H. and Porter, T.: A homotopy 2-groupoid from a fibration, Homotopy, Homology Appl. 1 (1999), 79-93.
Loday, J.-L.: Spaces with finitely many homotopy groups, J. Pure Appl. Algebra 24 (1982), 179-201.
Mackaay, M. and Picken, R.: Holonomy and parallel transport for abelian gerbes, Adv. Math. 170 (2002), 287-339.
Moerdijk, I. and Svensson, J.-A.: Algebraic classification of equivariant 2-types, J. Pure Appl. Algebra 89 (1993), 187-216.
Quillen, D.: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer, Berlin, 1967.
Spencer, C. B.: An abstract setting for homotopy pushouts and pullbacks, Cahiers Topologie Géom. Différentielle Catég. 18 (1977), 409-430.
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Brown, R., Janelidze, G. Galois Theory and a New Homotopy Double Groupoid of a Map of Spaces. Applied Categorical Structures 12, 63–80 (2004). https://doi.org/10.1023/B:APCS.0000013811.15727.1a
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DOI: https://doi.org/10.1023/B:APCS.0000013811.15727.1a