Some phenomena in homotopical algebra

Author:
K. Varadarajan

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

MSC:
Primary 55D40

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]**Albrecht Dold,*Partitions of unity in the theory of fibrations*, Ann. of Math. (2)**78**(1963), 223–255. MR**0155330****[2]**P. J. Hilton,*Homotopy theory of modules and duality*, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 273–281. MR**0098126****[3]**Peter Hilton,*Homotopy theory and duality*, Gordon and Breach Science Publishers, New York-London-Paris, 1965. MR**0198466****[4]**Michel A. Kervaire,*Smooth homology spheres and their fundamental groups*, Trans. Amer. Math. Soc.**144**(1969), 67–72. MR**0253347**, 10.1090/S0002-9947-1969-0253347-3**[5]**John Milnor,*Construction of universal bundles. I*, Ann. of Math. (2)**63**(1956), 272–284. MR**0077122****[6]**Daniel G. Quillen,*Homotopical algebra*, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR**0223432****[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]**K. Varadarajan,*Groups for which Moore spaces 𝑀(𝜋,1) exist*, Ann. of Math. (2)**84**(1966), 368–371. MR**0202143**

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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