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A model structure on the category of pro-simplicial sets


Author: Daniel C. Isaksen
Journal: Trans. Amer. Math. Soc. 353 (2001), 2805-2841
MSC (2000): Primary 18E35, 55Pxx, 55U35; Secondary 14F35, 55P60
DOI: https://doi.org/10.1090/S0002-9947-01-02722-2
Published electronically: January 29, 2001
MathSciNet review: 1828474
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Abstract:

We study the category ${pro-}\mathcal{SS}$ of pro-simplicial sets, which arises in étale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on ${pro-}\mathcal{SS}$ so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for ind-spaces.


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Additional Information

Daniel C. Isaksen
Affiliation: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
Address at time of publication: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: isaksen.1@nd.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02722-2
Keywords: Closed model structures, pro-spaces, \'etale homotopy
Received by editor(s): October 12, 1999
Published electronically: January 29, 2001
Additional Notes: The author was supported in part by an NSF Graduate Fellowship
Article copyright: © Copyright 2001 American Mathematical Society

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