Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2019, , 519 - 531, 01.08.2019
https://doi.org/10.16984/saufenbilder.472805

Öz

Kaynakça

  • H.F. Akız, N. Alemdar, O. Mucuk and T. Şahan, “Coverings of internal groupoids and crossed modules in the category of groups with operations,” Georgian Mathematical Journal, vol. 20, no. 2, pp. 223–238, 2013.
  • Z. Arvasi, “Crossed Squares and 2-Crossed Modules of Commutative Algebras”, Theory and Applications of Categories, vol. 3, no. 7, pp. 160--181, 1997.
  • Z. Arvasi and T. Porter, “Freeness Conditions for 2-Crossed Modules of Commutative Algebras,” Applied Categorical Structures, vol. 6, no. 4, pp. 455--471, 1988.
  • R. Brown, “Coproducts of crossed P-modules: Applications to second homotopy groups and to the homology of groups,” Topology, vol. 23, pp. 337--345, 1984.
  • R. Brown and P.J. Higgins, “On the connection between the second relative homotopy groups of some related spaces,” Proceedings of the London Mathematical Society, vol. 36, pp. 193--212, 1978.
  • R. Brown, R. and J. Huebschmann, “Identities among relations,” in Low Dimensional Topology, London Mathematical Society Lecture Note Series, vol. 48, pp. 153--202, 1982.
  • R. Brown and J.-L. Loday, “Homotopical excision, and Hurwicz theorems, for n-cubes of spaces,” Proceedings of the London Mathematical Society, vol. 54, no. 3, pp. 176--192, 1987.
  • R. Brown and C.B. Spencer, “G-groupoids, crossed modules and the fundamental groupoid of a topological group,” Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen series A, vol. 79, no. 4, pp. 296--302, 1976.
  • D. Conduché, “Modules croisés généralisés de longueur 2,” Journal of Pure and Applied Algebra, vol. 34, pp. 155--178, 1984.
  • G.J. Ellis, “Higher dimensional crossed modules of algebras,” Journal of Pure and Applied Algebra, vol. 52, pp. 277--282, 1988.
  • M. Gerstenhaber, “On the deformation of rings and algebras,” Annals of Mathematics, vol. 84, pp. 1--19, 1966.
  • D. Guin-Walery, D. and J.-L. Loday, “Obstructions Alexcision en K-theorie Algebrique,” Springer Lecture Notes in Mathematics, vol. 854, pp. 179--216, 1981.
  • S. Lichtenbaum and M. Schlessinger, "The cotangent complex of a morphism", Transactions of the American Mathematical Society, vol. 128, pp. 41--70, 1967.
  • J.-L. Loday, “Spaces with finitely many non-trivial homotopy groups,” Journal of Pure and Applied Algebra, vol. 24, pp. 179--202, 1982.
  • O. Mucuk and H.F. Akız, “Monodromy groupoids of an internal groupoid in topological groups with operations,” Filomat, vol. 29, no. 10, pp. 2355--2366, 2015.
  • O. Mucuk, B. Kılıçarslan, T. Şahan and N. Alemdar, “Group-groupoids and monodromy groupoids,” Topology and its Applications, vol. 158, no. 15, pp. 2034--2042, 2011.
  • O. Mucuk and T. Şahan, “Coverings and crossed modules of topological groups with operations,” Turkish Journal of Mathematics, vol. 38, no. 5, pp. 833--845, 2014.
  • O. Mucuk, T. Şahan and N. Alemdar, “Normality and Quotients in Crossed Modules and Group-groupoids,” Applied Categorical Structures, vol. 23, pp. 415--428, 2015.
  • A. Mutlu and T. Porter, “Freeness Conditions for 2-Crossed Modules and Complexes,” Applied Categorical Structures, vol. 4, no. 8, pp. 174--194, 1998.
  • A. Mutlu and T. Porter, “Freeness Conditions for Crossed Squares and Squared Complexes,” Kluwer Academic Publishers, vol. 20, no. 8, pp. 345--368, 2000.
  • K.J. Norrie, “Crossed modules and analogues of group theorems”, Ph. D. Thesis, King's College, England, UK, 1987.
  • K.J. Norrie, “Actions and automorphisms of crossed modules,” Bulletin de la Société Mathématique de France, vol. 118, no. 2, pp. 129--146, 1990.
  • G. Orzech, “Obstruction theory in categories. I - II,” Journal of Pure and Applied Algebra, vol. 2, pp. 287--314, 315--340, 1972.
  • T. Porter, “Extensions, crossed modules and internal categories in categories of groups with operations,” Proceedings of the Edinburgh Mathematical Society, vol. 30, pp. 373--381, 1987.
  • J.H.C. Whitehead, “Note on a previous paper entitled "On adding relations to homotopy group",” Annals of Mathematics, vol. 47, pp. 806--810, 1946.
  • J.H.C. Whitehead, “On operators in relative homotopy groups,” Annals of Mathematics, vol. 49, pp. 610--640, 1948.
  • J.H.C. Whitehead, “Combinatorial homotopy II,” Bulletin of the American Mathematical Society, vol. 55, pp. 453--496, 1949.

Categories internal to crossed modules

Yıl 2019, , 519 - 531, 01.08.2019
https://doi.org/10.16984/saufenbilder.472805

Öz

In
this study, internal categories in the category of the crossed modules are
characterized and it has been shown that there is a natural equivalence between
the category of crossed modules over crossed modules, i.e. crossed squares and
the category of the internal categories within the category of crossed modules.
Finally, we obtain examples of crossed squares using this equivalence.

Kaynakça

  • H.F. Akız, N. Alemdar, O. Mucuk and T. Şahan, “Coverings of internal groupoids and crossed modules in the category of groups with operations,” Georgian Mathematical Journal, vol. 20, no. 2, pp. 223–238, 2013.
  • Z. Arvasi, “Crossed Squares and 2-Crossed Modules of Commutative Algebras”, Theory and Applications of Categories, vol. 3, no. 7, pp. 160--181, 1997.
  • Z. Arvasi and T. Porter, “Freeness Conditions for 2-Crossed Modules of Commutative Algebras,” Applied Categorical Structures, vol. 6, no. 4, pp. 455--471, 1988.
  • R. Brown, “Coproducts of crossed P-modules: Applications to second homotopy groups and to the homology of groups,” Topology, vol. 23, pp. 337--345, 1984.
  • R. Brown and P.J. Higgins, “On the connection between the second relative homotopy groups of some related spaces,” Proceedings of the London Mathematical Society, vol. 36, pp. 193--212, 1978.
  • R. Brown, R. and J. Huebschmann, “Identities among relations,” in Low Dimensional Topology, London Mathematical Society Lecture Note Series, vol. 48, pp. 153--202, 1982.
  • R. Brown and J.-L. Loday, “Homotopical excision, and Hurwicz theorems, for n-cubes of spaces,” Proceedings of the London Mathematical Society, vol. 54, no. 3, pp. 176--192, 1987.
  • R. Brown and C.B. Spencer, “G-groupoids, crossed modules and the fundamental groupoid of a topological group,” Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen series A, vol. 79, no. 4, pp. 296--302, 1976.
  • D. Conduché, “Modules croisés généralisés de longueur 2,” Journal of Pure and Applied Algebra, vol. 34, pp. 155--178, 1984.
  • G.J. Ellis, “Higher dimensional crossed modules of algebras,” Journal of Pure and Applied Algebra, vol. 52, pp. 277--282, 1988.
  • M. Gerstenhaber, “On the deformation of rings and algebras,” Annals of Mathematics, vol. 84, pp. 1--19, 1966.
  • D. Guin-Walery, D. and J.-L. Loday, “Obstructions Alexcision en K-theorie Algebrique,” Springer Lecture Notes in Mathematics, vol. 854, pp. 179--216, 1981.
  • S. Lichtenbaum and M. Schlessinger, "The cotangent complex of a morphism", Transactions of the American Mathematical Society, vol. 128, pp. 41--70, 1967.
  • J.-L. Loday, “Spaces with finitely many non-trivial homotopy groups,” Journal of Pure and Applied Algebra, vol. 24, pp. 179--202, 1982.
  • O. Mucuk and H.F. Akız, “Monodromy groupoids of an internal groupoid in topological groups with operations,” Filomat, vol. 29, no. 10, pp. 2355--2366, 2015.
  • O. Mucuk, B. Kılıçarslan, T. Şahan and N. Alemdar, “Group-groupoids and monodromy groupoids,” Topology and its Applications, vol. 158, no. 15, pp. 2034--2042, 2011.
  • O. Mucuk and T. Şahan, “Coverings and crossed modules of topological groups with operations,” Turkish Journal of Mathematics, vol. 38, no. 5, pp. 833--845, 2014.
  • O. Mucuk, T. Şahan and N. Alemdar, “Normality and Quotients in Crossed Modules and Group-groupoids,” Applied Categorical Structures, vol. 23, pp. 415--428, 2015.
  • A. Mutlu and T. Porter, “Freeness Conditions for 2-Crossed Modules and Complexes,” Applied Categorical Structures, vol. 4, no. 8, pp. 174--194, 1998.
  • A. Mutlu and T. Porter, “Freeness Conditions for Crossed Squares and Squared Complexes,” Kluwer Academic Publishers, vol. 20, no. 8, pp. 345--368, 2000.
  • K.J. Norrie, “Crossed modules and analogues of group theorems”, Ph. D. Thesis, King's College, England, UK, 1987.
  • K.J. Norrie, “Actions and automorphisms of crossed modules,” Bulletin de la Société Mathématique de France, vol. 118, no. 2, pp. 129--146, 1990.
  • G. Orzech, “Obstruction theory in categories. I - II,” Journal of Pure and Applied Algebra, vol. 2, pp. 287--314, 315--340, 1972.
  • T. Porter, “Extensions, crossed modules and internal categories in categories of groups with operations,” Proceedings of the Edinburgh Mathematical Society, vol. 30, pp. 373--381, 1987.
  • J.H.C. Whitehead, “Note on a previous paper entitled "On adding relations to homotopy group",” Annals of Mathematics, vol. 47, pp. 806--810, 1946.
  • J.H.C. Whitehead, “On operators in relative homotopy groups,” Annals of Mathematics, vol. 49, pp. 610--640, 1948.
  • J.H.C. Whitehead, “Combinatorial homotopy II,” Bulletin of the American Mathematical Society, vol. 55, pp. 453--496, 1949.
Toplam 27 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Araştırma Makalesi
Yazarlar

Tunçar Şahan 0000-0002-6552-4695

Jihad Jamil Mohammed Bu kişi benim 0000-0001-5633-0971

Yayımlanma Tarihi 1 Ağustos 2019
Gönderilme Tarihi 20 Ekim 2018
Kabul Tarihi 20 Aralık 2018
Yayımlandığı Sayı Yıl 2019

Kaynak Göster

APA Şahan, T., & Mohammed, J. J. (2019). Categories internal to crossed modules. Sakarya University Journal of Science, 23(4), 519-531. https://doi.org/10.16984/saufenbilder.472805
AMA Şahan T, Mohammed JJ. Categories internal to crossed modules. SAUJS. Ağustos 2019;23(4):519-531. doi:10.16984/saufenbilder.472805
Chicago Şahan, Tunçar, ve Jihad Jamil Mohammed. “Categories Internal to Crossed Modules”. Sakarya University Journal of Science 23, sy. 4 (Ağustos 2019): 519-31. https://doi.org/10.16984/saufenbilder.472805.
EndNote Şahan T, Mohammed JJ (01 Ağustos 2019) Categories internal to crossed modules. Sakarya University Journal of Science 23 4 519–531.
IEEE T. Şahan ve J. J. Mohammed, “Categories internal to crossed modules”, SAUJS, c. 23, sy. 4, ss. 519–531, 2019, doi: 10.16984/saufenbilder.472805.
ISNAD Şahan, Tunçar - Mohammed, Jihad Jamil. “Categories Internal to Crossed Modules”. Sakarya University Journal of Science 23/4 (Ağustos 2019), 519-531. https://doi.org/10.16984/saufenbilder.472805.
JAMA Şahan T, Mohammed JJ. Categories internal to crossed modules. SAUJS. 2019;23:519–531.
MLA Şahan, Tunçar ve Jihad Jamil Mohammed. “Categories Internal to Crossed Modules”. Sakarya University Journal of Science, c. 23, sy. 4, 2019, ss. 519-31, doi:10.16984/saufenbilder.472805.
Vancouver Şahan T, Mohammed JJ. Categories internal to crossed modules. SAUJS. 2019;23(4):519-31.

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