Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Completions of pro-spaces   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 R = Z/p, the first homotopy
category is equivalent to a homotopy theory defined by Morel but has some
convenient categorical advantages.

24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
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The categories of this paper are: econ
20
The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol