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Completions of pro-spaces

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Abstract.

For every ring R, we present a pair of model structures on the category of pro-spaces. In the first, the weak equivalences are detected by cohomology with coefficients in R. In the second, the weak equivalences are detected by cohomology with coefficients in all R-modules (or equivalently by pro-homology with coefficients in R). In the second model structure, fibrant replacement is essentially just the Bousfield-Kan R-tower. When the first homotopy category is equivalent to a homotopy theory defined by Morel but has some convenient categorical advantages.

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Daniel C. Isaksen

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  1. Daniel C. Isaksen
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Mathematical Subject Classification (1991): 55P60, 55N10 18G55, 55U35

This work was partially supported by a National Science Foundation Postdoctoral Research Fellowship. The author acknowledges useful conversations with Bill Dwyer and Daniel Biss. The author thanks the referee for several corrections and excellent suggestions.

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Isaksen, D. Completions of pro-spaces. Math. Z. 250, 113–143 (2005). https://doi.org/10.1007/s00209-004-0745-x

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