Some phenomena in homotopical algebra

Author:
K. Varadarajan

Journal:
Proc. Amer. Math. Soc. **43** (1974), 272-276

MSC:
Primary 55D40

DOI:
https://doi.org/10.1090/S0002-9939-1974-0377870-3

MathSciNet review:
0377870

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Abstract: In [**6**] D. G. Quillen developed homotopy theory in categories satisfying certain axioms. He showed that many results in classical homotopy theory (of topological spaces) go through in his axiomatic set-up. The duality observed by Eckmann-Hilton in classical homotopy theory is reflected in the axioms of a model category. In [**7**] we developed the theory of numerical invariants like the Lusternik-Schnirelmann category and cocategory etc. for such model categories and in [**8**] we dealt with applications of this theory to injective and projective homotopy theory of modules as developed by Hilton [**2**], [**3**, Chapter 13].

Contrary to the general expectations there are many aspects of classical homotopy theory which cannot be carried over to Quillen's axiomatic set-up. This paper deals with some of these phenomena.

**[1]**A. Dold,*Partitions of unity in the theory of fibrations*, Ann. of Math. (2)**78**(1963), 223-255. MR**27**#5264. MR**0155330 (27:5264)****[2]**P. J. Hilton,*Homotopy theory of modules and duality*, Internat. Sympos. on Algebraic Topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 273-281. MR**20**#4588. MR**0098126 (20:4588)****[3]**-,*Homotopy theory and duality*, Gordon and Breach, New York, 1965. MR**33**#6624. MR**0198466 (33:6624)****[4]**M. A. Kervaire,*Smooth homology spheres and their fundamental groups*, Trans. Amer. Math. Soc.**144**(1969), 67-72. MR**40**#6562. MR**0253347 (40:6562)****[5]**J. W. Milnor,*Construction of universal bundles*. I, Ann. of Math. (2)**63**(1956), 272-284. MR**17**, 994. MR**0077122 (17:994b)****[6]**D. G. Quillen,*Homotopical algebra*, Lecture Notes in Math., no. 43, Springer-Verlag, Berlin and New York, 1967. MR**36**#6480. MR**0223432 (36:6480)****[7]**K. Varadarajan,*Numerical invariants in homotopical algebra*, J. Pure Appl. Algebra (submitted).**[8]**-,*Numerical invariants in homotopical algebra*. II,*Applications*, J. Pure Appl. Algebra (submitted).**[9]**-,*Groups for which Moore spaces exist*, Ann. of Math. (2)**84**(1966), pp. 368-371. MR**34**#2017. MR**0202143 (34:2017)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0377870-3

Keywords:
Category,
cocategory,
-homotopy,
-homotopy,
Moore spaces

Article copyright:
© Copyright 1974
American Mathematical Society