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Proceedings of the American Mathematical Society

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

 

 

Homotopy limits and the homotopy type of functor categories


Author: David A. Cox
Journal: Proc. Amer. Math. Soc. 58 (1976), 55-58
MSC: Primary 14F35; Secondary 18A25
DOI: https://doi.org/10.1090/S0002-9939-1976-0407022-1
MathSciNet review: 0407022
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Abstract: Let $ {\mathbf{Y}}:I \to $ Simplicial Sets be a functor. We give a sufficient condition for the map $ \mathop {{\text{ho}}\lim }\limits_ \to {\mathbf{Y}} \to \mathop {\lim }\limits_ \to {\mathbf{Y}}$ to be a weak equivalence. Then we apply this to determine the Artin-Mazur homotopy type of the functor category Funct($ I$, Sets).


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

  • [1] Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354653
  • [2] M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin-New York, 1969. MR 0245577
  • [3] 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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0407022-1
Keywords: Simplicial set, homotopy direct limit, functor category, Artin-Mazur homotopy type, hypercovering, nerve of a category
Article copyright: © Copyright 1976 American Mathematical Society