Crossed Cat1-Modules

Yıl 2023, Cilt: 13 Sayı: 4, 2958 - 2972, 01.12.2023

Tunçar Şahan , Emre Kendir

https://doi.org/10.21597/jist.1303212

Öz

It is well known that crossed modules over groups are an algebraic model of homotopy 2-type connected spaces. Moreover, cat1-groups and internal categories in the category of groups, i.e. 2-groups or group-groupoids, are categorically equivalent to crossed modules over groups. In this study, as a new algebraic model of homotopy 3-type connected spaces, the algebraic structure of the crossed module on the category of cat1-groups, i.e. the crossed cat1-module, is characterized and some of its properties are studied. It is also shown that crossed cat1-modules are categorically equivalent to crossed squares over groups and hence to cat2-groups.

Anahtar Kelimeler

Crossed modüle, Cat1-group, Crossed square, Cat2-group

Çaprazlanmış Cat1-Modüller

Öz

Gruplar üzerindeki çaprazlanmış modüllerin homotopi 2-tipten bağlantılı uzayların bir cebirsel modeli olduğu iyi bilinen bir gerçektir. Ayrıca cat1-gruplar ve grupların kategorisindeki iç kategoriler, diğer bir ifadeyle 2-gruplar veya grup-grupoidler, kategoriksel olarak gruplar üzerindeki çaprazlanmış modüllere denktirler. Bu çalışmada, homotopi 3-tipten bağlantılı uzayların yeni bir cebirsel modeli olarak cat1-grupların kategorisindeki çaprazlanmış modül, yani çaprazlanmış cat1-modül, cebirsel yapısı karakterize edilip bazı özellikleri incelenmiştir. Ayrıca çaprazlanmış cat1-modüllerin kategoriksel olarak gruplar üzerindeki çaprazlanmış karelere ve böylece cat2-gruplara denk oldukları gösterilmiştir.

Anahtar Kelimeler

Çaprazlanmış modül, Cat1-grup, Çaprazlanmış kare, Cat2-grup

Kaynakça

• Akız, H. F., Alemdar, N., Mucuk, O. ve Şahan, T. (2013). Coverings of internal groupoids and crossed modules in the category of groups with operations. Georgian Mathematical Journal, 20(2), 223 – 238.
• Akız, H. F., Mucuk, O. ve Şahan, T. (2020). Liftings of crossed modules in the category of groups with operations. Boletim da Sociedade Paranaense de Matemática, 38(7), 181 – 193.
• Alp, M. (1998). Pullbacks of crossed modules and cat1-groups. Turkish Journal of Mathematics, 22, 273 – 281.
• Arvasi, Z. (1997). Crossed squares and 2-crossed modules of commutative algebras. Theory and Applications of Categories, 3(7), 160 – 181.
• Brown, R. (1984). Coproducts of crossed P-modules: Applications to second homotopy groups and to the homology of groups. Topology, 23(3), 337 – 345.
• Brown, R. (1987). From groups to groupoids: a brief survey. Bulletin of the London Mathematical Society, 19, 113 – 134.
• Brown, R. ve Higgins, P. J. (1978). On the connection between the second relative homotopy groups of some related spaces. Proceedings of the London Mathematical Society, 36, 193-212.
• Brown, R. ve Higgins, P. J. (1991). The classifying space of a crossed complex. Mathematical Proceedings of the Cambridge Philosophical Society, 110(1), 95 – 120.
• Brown, R. ve Loday, J. L. (1987a). Homotopical excision, and Hurwicz theorems, for n-cubes of spaces. Proceedings of the London Mathematical Society, 54(3), 176 – 192.
• Brown, R. ve Loday, J. L. (1987b). Van Kampen theorems for diagrams of spaces. Topology, 26(3), 311 – 335.
• Brown, R. ve Spencer, C. B. (1976). G-groupoids, crossed modules and the fundamental groupoid of a topological group. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen series A, 79(4), 296 – 302.
• Conduché, D. (1984). Modules croisés généralisés de longueur 2. Journal of Pure and Applied Algebra, 34(2-3), 155 – 178.
• Dijkgraaf, R. ve Witten, E. (1990). Topological gauge theories and group cohomology. Communications in Mathematical Physics, 129(2), 393 – 429.
• Ellis, G. J. (1988). Higher dimensional crossed modules of algebras. Journal of Pure and Applied Algebra, 52, 277 – 282.
• Higgins, P. J. (1956). Groups with multiple operators. Proceedings of the London Mathematical Society, 3(6), 366 – 416.
• Jurčo, B. (2011). Crossed module bundle gerbes; classification, string group and differential geometry. International Journal of Geometric Methods in Modern Physics, 8(05), 1079 – 1095.
• Loday, J. L. (1982). Spaces with finitely many non-trivial homotopy groups. Journal of Pure and Applied Algebra, 24, 179 – 202.
• Mac Lane, S. ve Whitehead, J. H. C. (1950). On the 3-types of a complex. Proceedings of the National Academy of Sciences, 36(1), 41 – 48.
• Mackenzie, K. C. H. (1987). Lie groupoids and Lie algebroids in differential geometry. London Mathematical Society Lecture Note Series 124, Cambridge University Press.
• Martins, J. F. ve Picken, R. (2011). The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module. Differential Geometry and its Applications, 29(2), 179 – 206.
• Mucuk, O. ve Şahan, T. (2014). Coverings and crossed modules of topological groups with operations. Turkish Journal of Mathematics, 38(5), 833 – 845.
• Mucuk, O. ve Şahan, T. (2019). Group-groupoid actions and liftings of crossed modules. Georgian Mathematical Journal, 26(3), 437 – 447.
• Mucuk, O., Şahan, T. ve Alemdar, N. (2014). Normality and quotients in crossed modules and group-groupoids. Applied Categorical Structures, 23, 415 – 428.
• Mutlu, A. ve Porter, T. (2000). Freeness conditions for crossed squares and squared complexes, Kluwer Academic Publishers, 20(8), 345 – 368.
• Norrie, K. (1987). Crossed modules and analogues of groups theorems. Dissertation, King’s College, University of London.
• Norrie, K. (1990). Actions and automorphisms of crossed modules. Bulletin de la Société Mathématique de France, 118(2), 129 – 146.
• Orzech, G. (1972a). Obstruction theory in categories. I. Journal of Pure and Applied Algebra, 2, 287 – 314.
• Orzech, G. (1972b). Obstruction theory in categories. II. Journal of Pure and Applied Algebra, 2, 315 – 340.
• Porter, T. (1987). Extensions, crossed modules and internal categories in categories of groups with operations. Proceedings of the Edinburgh Mathematical Society, 30, 373 – 381.
• Porter, T. (1998). Topological quantum field theories from homotopy n-types. Journal of the London Mathematical Society, 58(3), 723 – 732.
• Şahan, T. (2019). Further remarks on liftings of crossed modules. Hacettepe Journal of Mathematics and Statistics, 48(3), 743 – 752.
• Şahan, T. ve Mucuk, O. (2020). Normality and quotient in the category of crossed modules within the category of groups with operations. Boletim da Sociedade Paranaense de Matemática, 38(7), 169 – 179.
• Whitehead, J.H.C. (1946). Note on a previous paper entitled "On adding relations to homotopy groups. Annals of Mathematics, 47(4), 806 – 810.
• Whitehead, J.H.C. (1948). On operators in relative homotopy groups. Annals of Mathematics, 49, 610 – 640.
• Whitehead, J.H.C. (1949). Combinatorial homotopy. II. Bulletin of the American Mathematical Society, 55, 213 – 245.
• Yetter, D.N. (1992). Topological quantum field theories associated to finite groups and crossed G-sets. Journal of Knot Theory and Its Ramifications, 1, 1–20.
• Yetter, D.N. (1993). TQFT’s from homotopy 2-types. Journal of Knot Theory and Its Ramifications, 2, 113 – 123.