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Certain geometric properties of covariant functors. (English. Russian original) Zbl 0572.54013

Russ. Math. Surv. 39, No. 5, 199-249 (1984); translation from Usp. Mat. Nauk 39, No. 5(239), 169-208 (1984).
The title of the paper is somewhat misleading insofar as the author studies properties of geometric functors rather than geometric properties of functors. The meaning and usage of ”geometric functor” is best illustrated by three examples. The first functor associates with each separated topological space \(X\) the space \(\exp X\) of all non-empty compact subsets of \(X\); the second functor associates with \(X\) the space \(P(X)\) of all probability laws on \(X\); the third functor associates with \(X\) the space \(\Gamma(X)\) of all oriented arcs of \(\exp X\). The author discusses (mostly without proofs) geometric functors which preserve certain properties of topological spaces (such as being locally connected or an absolute retract or a Peano continuum) or which preserve certain properties of continuous mappings (such as having absolute retracts as fibres).
Reviewer: J.Sonner

MSC:

54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54B20 Hyperspaces in general topology
54B30 Categorical methods in general topology
18B30 Categories of topological spaces and continuous mappings (MSC2010)
54F15 Continua and generalizations
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