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Cofibration Category and Homotopies of Three–Crossed Complexes

Year 2013, Volume: 42 Issue: 1, 31 - 41, 01.01.2013

Abstract

In this work, we show that category of totally free 2–crossed complexes and that of totally free 3–crossed complexes are cofibration categories in the sense of Baues ([4]). We also explore homotopies for 3–crossed modules and 3–crossed complex morphisms.

References

  • Arvasi, Z. and Ulualan, E. Quadratic and 2–crossed modules of algebras, Algebra Collo- quium, 14 No. 4, 669–686, 2007.
  • Arvasi, Z., Kuzpinari, T. S. and Uslu, E. ¨O. Three–crossed Modules, Homology Homotopy Appl., 11, 161–187, 2009.
  • Baues, H. J. Combinatorial homotopy and 4–dimensional complexes, (Walter de Gruyter, 1991).
  • Baues, H. J. Algebraic homotopy, (Cambridge Studies in Advanced Mathematics, 1998).
  • Brown, R. and Higgins, P. J. The classifying space of a crossed complex, Math. Proc. Cam- bridge Phil. Soc., 110, 95–120, 1991.
  • Brown R. and Higgins, P. J. Tensor Products and Homotopies for ω–groupoids and Crossed Complexes, J.P.A.A 47, 11–44, 1987.
  • Brown, R. and Golanski, M. A model structure for the homotopy theory of crossed complexes, Cah. Top. G´eom. Diff. Cat, 30, 61–82, 1989.
  • Brown, R. and ˙I¸cen, ˙I. Homotopies and automorphisms of crossed modules over groupoids, Appl. Categorical Structure, 11, 185–206, 2003.
  • Conduch´e, D. Modules crois´es g´e n´eralis´es de longueur 2, Journal of Pure and Applied Algebra, 34, 155–178, 1984.
  • Kamps, K. H. Kan–Bedingungen und abstrakte Homotopie theorie, Math. Z., 124, 215–236, 19 Martin, Joao Faria Homotopies of 2–crossed complexes and the homotopy category of pointed 3–types, http://arxiv.org/pdf/math/0605364v1.pdf, 2011.
  • Mutlu, A. and Porter, T. Freeness conditions for 2–crossed modules and complexes, Theory and Applications of Categories, 4 No. 8, 174–194, 1998.
  • Quillen D. Lecture Notes in Math., Homotopical Algebra, 11, 185–206, 1967.
  • Radulescu–Banu, Andrei Cofibrations in Homotopy Theory, http://arxiv.org/abs/math/ 0610009v4, 2009.
  • Whitehead, J. H. C. Combinatorial homotopy II, Bull. Amer. Math. Soc., 55, 453–496, 1949.

Cofibration Category and Homotopies of Three–Crossed Complexes

Year 2013, Volume: 42 Issue: 1, 31 - 41, 01.01.2013

Abstract

References

  • Arvasi, Z. and Ulualan, E. Quadratic and 2–crossed modules of algebras, Algebra Collo- quium, 14 No. 4, 669–686, 2007.
  • Arvasi, Z., Kuzpinari, T. S. and Uslu, E. ¨O. Three–crossed Modules, Homology Homotopy Appl., 11, 161–187, 2009.
  • Baues, H. J. Combinatorial homotopy and 4–dimensional complexes, (Walter de Gruyter, 1991).
  • Baues, H. J. Algebraic homotopy, (Cambridge Studies in Advanced Mathematics, 1998).
  • Brown, R. and Higgins, P. J. The classifying space of a crossed complex, Math. Proc. Cam- bridge Phil. Soc., 110, 95–120, 1991.
  • Brown R. and Higgins, P. J. Tensor Products and Homotopies for ω–groupoids and Crossed Complexes, J.P.A.A 47, 11–44, 1987.
  • Brown, R. and Golanski, M. A model structure for the homotopy theory of crossed complexes, Cah. Top. G´eom. Diff. Cat, 30, 61–82, 1989.
  • Brown, R. and ˙I¸cen, ˙I. Homotopies and automorphisms of crossed modules over groupoids, Appl. Categorical Structure, 11, 185–206, 2003.
  • Conduch´e, D. Modules crois´es g´e n´eralis´es de longueur 2, Journal of Pure and Applied Algebra, 34, 155–178, 1984.
  • Kamps, K. H. Kan–Bedingungen und abstrakte Homotopie theorie, Math. Z., 124, 215–236, 19 Martin, Joao Faria Homotopies of 2–crossed complexes and the homotopy category of pointed 3–types, http://arxiv.org/pdf/math/0605364v1.pdf, 2011.
  • Mutlu, A. and Porter, T. Freeness conditions for 2–crossed modules and complexes, Theory and Applications of Categories, 4 No. 8, 174–194, 1998.
  • Quillen D. Lecture Notes in Math., Homotopical Algebra, 11, 185–206, 1967.
  • Radulescu–Banu, Andrei Cofibrations in Homotopy Theory, http://arxiv.org/abs/math/ 0610009v4, 2009.
  • Whitehead, J. H. C. Combinatorial homotopy II, Bull. Amer. Math. Soc., 55, 453–496, 1949.
There are 14 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Mathematics
Authors

Hasan Atik This is me

Publication Date January 1, 2013
Published in Issue Year 2013 Volume: 42 Issue: 1

Cite

APA Atik, H. (2013). Cofibration Category and Homotopies of Three–Crossed Complexes. Hacettepe Journal of Mathematics and Statistics, 42(1), 31-41.
AMA Atik H. Cofibration Category and Homotopies of Three–Crossed Complexes. Hacettepe Journal of Mathematics and Statistics. January 2013;42(1):31-41.
Chicago Atik, Hasan. “Cofibration Category and Homotopies of Three–Crossed Complexes”. Hacettepe Journal of Mathematics and Statistics 42, no. 1 (January 2013): 31-41.
EndNote Atik H (January 1, 2013) Cofibration Category and Homotopies of Three–Crossed Complexes. Hacettepe Journal of Mathematics and Statistics 42 1 31–41.
IEEE H. Atik, “Cofibration Category and Homotopies of Three–Crossed Complexes”, Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 1, pp. 31–41, 2013.
ISNAD Atik, Hasan. “Cofibration Category and Homotopies of Three–Crossed Complexes”. Hacettepe Journal of Mathematics and Statistics 42/1 (January 2013), 31-41.
JAMA Atik H. Cofibration Category and Homotopies of Three–Crossed Complexes. Hacettepe Journal of Mathematics and Statistics. 2013;42:31–41.
MLA Atik, Hasan. “Cofibration Category and Homotopies of Three–Crossed Complexes”. Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 1, 2013, pp. 31-41.
Vancouver Atik H. Cofibration Category and Homotopies of Three–Crossed Complexes. Hacettepe Journal of Mathematics and Statistics. 2013;42(1):31-4.