June  2022, 14(2): 349-375. doi: 10.3934/jgm.2022011

Towards a 2-dimensional notion of holonomy

1. 

Mathematics Division, School of Informatics, University of Wales, Bangor, Gwynedd LL57 1UT, UK

2. 

Department of Mathematics, Faculty of Science and Art, University of İnönü, Malatya, Turkey

Received  May 2022 Published  June 2022 Early access  June 2022

Fund Project: This article was published in Advances in Mathematics, 178, Ronald Brown and İlhan İçen "Towards a 2-dimensional notion of holonomy", 141–175, Copyright Elsevier (2003)

Previous work (Pradines, C.R. Acad. Sci. Paris 263 (1966) 907, Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the $ 2 $-dimensional information.

Citation: Ronald Brown, İlhan İçen. Towards a 2-dimensional notion of holonomy. Journal of Geometric Mechanics, 2022, 14 (2) : 349-375. doi: 10.3934/jgm.2022011
References:
[1]

M. A. E.-S. A.-F. Aof and R. Brown, The holonomy groupoid of a locally topological groupoid, Top. and its Appl., 47 (1992), 97-113.  doi: 10.1016/0166-8641(92)90065-8.

[2]

R. Brown, Holonomy and monodromy groupoids, U. C. N. W. Maths Preprint, (1982), 82.2.

[3]

R. Brown, Groupoids and crossed objects in algebraic topology, Homology, Homotopy and its Applications, 1 (1999), 1-78.  doi: 10.4310/HHA.1999.v1.n1.a1.

[4]

R. Brown and P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3), 36 (1978), 193-212.  doi: 10.1112/plms/s3-36.2.193.

[5]

R. Brown and P. J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra, 21 (1981), 233-260.  doi: 10.1016/0022-4049(81)90018-9.

[6]

R. Brown and P. J. Higgins, Tensor products and homotopies for $\omega$-groupoids and crossed complexes, J. Pure and Appl. Algebra, 47 (1987), 1-33.  doi: 10.1016/0022-4049(87)90099-5.

[7]

R. Brown and İ. İçen, Lie local subgroupoids and their holonomy and monodromy Lie groupoids, Top and its Appl., (to appear). doi: 10.1016/S0166-8641(00)00062-6.

[8]

R. Brown and İ. İçen, Homotopies and automorphism of crossed modules of groupoids, (submitted). doi: 10.1023/A:1023544303612.

[9]

R. Brown and K. C. H. Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra, 80 (1992), 237-272.  doi: 10.1016/0022-4049(92)90145-6.

[10]

R. Brown and G. H. Mosa, Double categories, $2$-categories, thin structures and connections, Theory Appl. Categ., 5 (1999), 163-175. 

[11]

R. Brown and O. Mucuk, The monodromy groupoid of a Lie groupoid, Cahier Top. Géom. Diff. Cat., 36 (1995), 345-369. 

[12]

R. Brown and O. Mucuk, Foliations, locally Lie groupoids and holonomy, Cahier Top. Géom. Diff. Cat, 37 (1995), 61-71. 

[13]

R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Topologie Géom. Differentielle, 17 (1976), 343-362. 

[14]

C. Ehresmann,, Structures feuilletées, Proc. 5th Conf. Canadian Math. Congress Montreal 1961, Oeuvres Compl etes II-2,563–624.

[15]

A. Ehresmann and C. Ehresmann, Multiple functors. IV: Monoidal closed structure on $Cat_n$, Cahiers Top. Géom. Différentielle, 20 (1979), 59-104. 

[16]

C. Ehresmann, Catégories structurées, Ann. Sci. École Norm. Sup., 80 (1963), 349-426. 

[17]

A. Haefliger, Groupoïdes d'holonomie et classifiant, Astérisque, 116 (1984), 70-97. 

[18]

J. Kubarski,, Local and nice structures of the groupoid of equivalence relation.,

[19]

K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society, Lecture Note Series, 124, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511661839.

[20]

K. C. H. Mackenzie, Double Lie algebroids and second order geometry, Advances in Mathematics, 96 (1992), 180-239.  doi: 10.1016/0001-8708(92)90036-K.

[21]

J. Phillips, The holonomic imperative and the homotopy groupoid of foliated manifolds, Rocky Mountain Journal of Maths, 17 (1987), 151-165.  doi: 10.1216/RMJ-1987-17-1-151.

[22]

J. Pradines, Théorie de Lie pour les groupoides différentiables, Relation entre propriétes locales et globales, C. R. Acad. Sci. Paris, 263 (1966), 907-910. 

[23]

J. Pradines, Géometrie différentielle au-dessus d'un groupoïde, C. R. Acad. sci. Paris, Sér. A, 266 (1967), 1194-1196. 

[24]

J. Pradines,, Fibres vectoriels doubles et calculs des jets non holonomes, Notes Polycopiées, (1974).

[25]

J. H. C. Whitehead, On operators in relative homotopy groups, Ann. of Math., 49 (1948), 610-640.  doi: 10.2307/1969048.

[26]

J. H. C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc., 55 (1949), 453-996.  doi: 10.1090/S0002-9904-1949-09213-3.

[27]

H. E. Winkelnkemper, The graph of a foliation, Ann. Global. Anal. Geom., 1 (1985), 51–75. doi: 10.1007/BF02329732.

show all references

References:
[1]

M. A. E.-S. A.-F. Aof and R. Brown, The holonomy groupoid of a locally topological groupoid, Top. and its Appl., 47 (1992), 97-113.  doi: 10.1016/0166-8641(92)90065-8.

[2]

R. Brown, Holonomy and monodromy groupoids, U. C. N. W. Maths Preprint, (1982), 82.2.

[3]

R. Brown, Groupoids and crossed objects in algebraic topology, Homology, Homotopy and its Applications, 1 (1999), 1-78.  doi: 10.4310/HHA.1999.v1.n1.a1.

[4]

R. Brown and P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3), 36 (1978), 193-212.  doi: 10.1112/plms/s3-36.2.193.

[5]

R. Brown and P. J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra, 21 (1981), 233-260.  doi: 10.1016/0022-4049(81)90018-9.

[6]

R. Brown and P. J. Higgins, Tensor products and homotopies for $\omega$-groupoids and crossed complexes, J. Pure and Appl. Algebra, 47 (1987), 1-33.  doi: 10.1016/0022-4049(87)90099-5.

[7]

R. Brown and İ. İçen, Lie local subgroupoids and their holonomy and monodromy Lie groupoids, Top and its Appl., (to appear). doi: 10.1016/S0166-8641(00)00062-6.

[8]

R. Brown and İ. İçen, Homotopies and automorphism of crossed modules of groupoids, (submitted). doi: 10.1023/A:1023544303612.

[9]

R. Brown and K. C. H. Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra, 80 (1992), 237-272.  doi: 10.1016/0022-4049(92)90145-6.

[10]

R. Brown and G. H. Mosa, Double categories, $2$-categories, thin structures and connections, Theory Appl. Categ., 5 (1999), 163-175. 

[11]

R. Brown and O. Mucuk, The monodromy groupoid of a Lie groupoid, Cahier Top. Géom. Diff. Cat., 36 (1995), 345-369. 

[12]

R. Brown and O. Mucuk, Foliations, locally Lie groupoids and holonomy, Cahier Top. Géom. Diff. Cat, 37 (1995), 61-71. 

[13]

R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Topologie Géom. Differentielle, 17 (1976), 343-362. 

[14]

C. Ehresmann,, Structures feuilletées, Proc. 5th Conf. Canadian Math. Congress Montreal 1961, Oeuvres Compl etes II-2,563–624.

[15]

A. Ehresmann and C. Ehresmann, Multiple functors. IV: Monoidal closed structure on $Cat_n$, Cahiers Top. Géom. Différentielle, 20 (1979), 59-104. 

[16]

C. Ehresmann, Catégories structurées, Ann. Sci. École Norm. Sup., 80 (1963), 349-426. 

[17]

A. Haefliger, Groupoïdes d'holonomie et classifiant, Astérisque, 116 (1984), 70-97. 

[18]

J. Kubarski,, Local and nice structures of the groupoid of equivalence relation.,

[19]

K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society, Lecture Note Series, 124, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511661839.

[20]

K. C. H. Mackenzie, Double Lie algebroids and second order geometry, Advances in Mathematics, 96 (1992), 180-239.  doi: 10.1016/0001-8708(92)90036-K.

[21]

J. Phillips, The holonomic imperative and the homotopy groupoid of foliated manifolds, Rocky Mountain Journal of Maths, 17 (1987), 151-165.  doi: 10.1216/RMJ-1987-17-1-151.

[22]

J. Pradines, Théorie de Lie pour les groupoides différentiables, Relation entre propriétes locales et globales, C. R. Acad. Sci. Paris, 263 (1966), 907-910. 

[23]

J. Pradines, Géometrie différentielle au-dessus d'un groupoïde, C. R. Acad. sci. Paris, Sér. A, 266 (1967), 1194-1196. 

[24]

J. Pradines,, Fibres vectoriels doubles et calculs des jets non holonomes, Notes Polycopiées, (1974).

[25]

J. H. C. Whitehead, On operators in relative homotopy groups, Ann. of Math., 49 (1948), 610-640.  doi: 10.2307/1969048.

[26]

J. H. C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc., 55 (1949), 453-996.  doi: 10.1090/S0002-9904-1949-09213-3.

[27]

H. E. Winkelnkemper, The graph of a foliation, Ann. Global. Anal. Geom., 1 (1985), 51–75. doi: 10.1007/BF02329732.

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