Proceedings of the American Mathematical Society

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



On topological methods in homological algebra

Authors: David A. Edwards and Harold M. Hastings
Journal: Proc. Amer. Math. Soc. 59 (1976), 389-393
MSC: Primary 18G25; Secondary 55J99
MathSciNet review: 0424906
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give an appropriate extension of the concept of ``tower of surjections'' to arbitrary inverse systems. We introduce a natural closed model structure (in the sense of D. Quillen) on the category of pro-(Simplicial Abelian Groups) and interpret our condition as the definition of fibrant object.

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

  • [1] M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin-New York, 1969. MR 0245577
  • [2] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
  • [3] David A. Edwards, Étale homotopy theory and shape, Algebraic and geometrical methods in topology (Conf. Topological Methods in Algebraic Topology, State Univ. New York, Binghamton, N.Y., 1973), Springer, Berlin, 1974, pp. 58–107. Lecture Notes in Math., Vol. 428. MR 0375295
  • [4] David A. Edwards and Ross Geoghegan, Stability theorems in shape and pro-homotopy, Trans. Amer. Math. Soc. 222 (1976), 389–403. MR 0423347, 10.1090/S0002-9947-1976-0423347-2
  • [5] David A. Edwards and Harold M. Hastings, Why the 𝑅-completion works, General Topology and Appl. 7 (1977), no. 2, 179–184. NSF-CBMS Regional Conference in Topology (Univ. Tennessee, Knoxville, Tenn., 1974). MR 0454966
  • [6] -, On homotopy inverse limits and the vanishing of $ {\lim ^s}$, S.U.N.Y. at Binghamton (preprint).
  • [7] -, Čech and Steenrod homotopy theory, with applications to geometric topology, Lecture Notes in Math., Springer-Verlag, Berlin and New York (to appear).
  • [8] A. Grothendieck, Technique de descente et théorèmes d'existence en géométrie algébrique, Séminaire Bourbaki, 1959-1960, Exposés 190, 195 (mimeographed notes).
  • [9] H. M. Hastings, Homotopy theory of pro-spaces. I, II, S.U.N.Y. at Binghamton, 1974 (preprint).
  • [10] C. U. Jensen, Les foncteurs dérivés de \varprojlim et leurs applications en théorie des modules, Lecture Notes in Mathematics, Vol. 254, Springer-Verlag, Berlin-New York, 1972. MR 0407091
  • [11] James Keesling, The Čech homology of compact connected abelian topological groups with applications to shape theory, Geometric topology (Proc. Conf., Park City, Utah, 1974) Springer, Berlin, 1975, pp. 325–331. Lecture Notes in Math., Vol. 438. MR 0405344
  • [12] James Keesling, On the Whitehead theorem in shape theory, Fund. Math. 92 (1976), no. 3, 247–253. MR 0431156
  • [13] J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
  • [14] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
  • [15] Jan-Erik Roos, Sur les foncteurs dérivés de \underleftarrowlim. Applications, C. R. Acad. Sci. Paris 252 (1961), 3702–3704 (French). MR 0132091
  • [16] Jean-Louis Verdier, Équivalence essentielle des systèmes projectifs, C. R. Acad. Sci. Paris 261 (1965), 4950–4953 (French). MR 0190209
  • [17] Z. Z. Yeh, Thesis, Princeton, N. J., 1959.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 18G25, 55J99

Retrieve articles in all journals with MSC: 18G25, 55J99

Additional Information

Keywords: Pro-(abelian groups), pro-homotopy, limit, $ {\lim ^s}$, closed model category
Article copyright: © Copyright 1976 American Mathematical Society