Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Homotopy for functors


Author: Ming Jung Lee
Journal: Proc. Amer. Math. Soc. 36 (1972), 571-577
MSC: Primary 55J10
DOI: https://doi.org/10.1090/S0002-9939-1972-0334212-5
Erratum: Proc. Amer. Math. Soc. 42 (1974), 648-650.
MathSciNet review: 0334212
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that natural transformations play the role of homotopy for (covariant) functors. Homotopic functors are shown to induce identical maps between the homology groups of categories. For a space X, there is an associated category $ \Lambda S(X)$. We show that the classifying space of $ \Lambda S(X)$ has the same homotopy type as X if X is a CW complex. Moreover, we prove that, for CW complexes X and Y, f and $ g:X \to Y$ are homotopic if and only if $ \Lambda S(f)$ and $ \Lambda S(g)$ are.


References [Enhancements On Off] (What's this?)

  • [1] D. W. Anderson, Simplicial K-theory and generalized homology theories. I (preprint).
  • [2] D. M. Kan, On c.s.s. complexes, Amer. J. Math. 79 (1957), 449-476. MR 19, 759. MR 0090047 (19:759e)
  • [3] M. J. Lee, A generalized Mayer-Vietoris sequence (submitted).
  • [4] -, Homology of small categories and its applications, Thesis, University of Rochester, 1971.
  • [5] J. P. May, Simplicial objects in algebraic topology, Van Nostrand Math. Studies, no. 11, Van Nostrand, Princeton, N.J., 1967. MR 36 #5942. MR 0222892 (36:5942)
  • [6] S. Okamoto, Partially ordered sets and semi-simplicial complexes, Math. Japon. 15 (1970), 69-79. MR 0292064 (45:1152a)
  • [7] G. Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. No. 34 (1968), 105-112. MR 38 #718. MR 0232393 (38:718)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55J10

Retrieve articles in all journals with MSC: 55J10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0334212-5
Keywords: Small category, simplicial set, homotopy, subdivision, morphism complex, classifying space
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society