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Calculating limits and colimits in pro-categories. (English) Zbl 1039.18002

The use of procategories was pioneered by Grothendieck in work on descent in algebraic geometry. This extended earlier ideas from algebraic topology on inverse systems and morphisms between them, especially with regard to Čech homology and thus to limits. Procategories have also been used in shape theory (geometric topology) and other areas of algebraic topology.
The proofs of completeness and cocompleteness properties in procategories are technically complex, yet calculations with limits and colimits in these situations are important for applications. In this paper, the author revisits these results and reveals some new and surprising results that would seem to be of importance for future applications

MSC:

18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
18Exx Categorical algebra
55U35 Abstract and axiomatic homotopy theory in algebraic topology
14F35 Homotopy theory and fundamental groups in algebraic geometry
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