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Cofibration Category and Homotopies of Three–Crossed Complexes

Year 2013, Volume: 42 Issue: 1 , 31 - 41 , 01.01.2013
https://izlik.org/JA39AS58JJ

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
https://izlik.org/JA39AS58JJ

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 Research Article
Authors

Hasan Atik This is me

Publication Date January 1, 2013
IZ https://izlik.org/JA39AS58JJ
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. https://izlik.org/JA39AS58JJ
AMA 1.Atik H. Cofibration Category and Homotopies of Three–Crossed Complexes. Hacettepe Journal of Mathematics and Statistics. 2013;42(1):31-41. https://izlik.org/JA39AS58JJ
Chicago Atik, Hasan. 2013. “Cofibration Category and Homotopies of Three–Crossed Complexes”. Hacettepe Journal of Mathematics and Statistics 42 (1): 31-41. https://izlik.org/JA39AS58JJ.
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 [1]H. Atik, “Cofibration Category and Homotopies of Three–Crossed Complexes”, Hacettepe Journal of Mathematics and Statistics, vol. 42, no. 1, pp. 31–41, Jan. 2013, [Online]. Available: https://izlik.org/JA39AS58JJ
ISNAD Atik, Hasan. “Cofibration Category and Homotopies of Three–Crossed Complexes”. Hacettepe Journal of Mathematics and Statistics 42/1 (January 1, 2013): 31-41. https://izlik.org/JA39AS58JJ.
JAMA 1.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, Jan. 2013, pp. 31-41, https://izlik.org/JA39AS58JJ.
Vancouver 1.Hasan Atik. Cofibration Category and Homotopies of Three–Crossed Complexes. Hacettepe Journal of Mathematics and Statistics [Internet]. 2013 Jan. 1;42(1):31-4. Available from: https://izlik.org/JA39AS58JJ