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Open maps, colimits, and a convenient category of fibre spaces. (English) Zbl 0559.18005

The mathematical usefulness of complete and cocomplete Cartesian closed categories is well established. Several such subcategories of the category of topological spaces and continuous functions have received extensive study, however when dealing with categories of spaces over a fixed base space, the situation is less tractable. Insistence upon completeness and cocompleteness may rule out the existence of certain exponents or vice versa. The paper under review investigates this problem with the key idea being the need for considering only open mappings. [The connection between openness and exponentiation has also been studied by P. T. Johnstone, Contemp. Math. 30, 84-116 (1984; Zbl 0537.18001).]
The author obtains the following results. Let \({\mathcal U}\) denote the category of compactly generated spaces and let \(B\in {\mathcal U}\). If \(Y\to^{q}B\) is an open map in \({\mathcal U}\), then pulling back along q preserves all colimits. Furthermore, if \({\mathcal O}(B)\) denotes the subcategory of \({\mathcal U}/B\) consisting of open maps, then \({\mathcal O}(B)\) is a complete and cocomplete Cartesian closed category. The author then proceeds to consider this construction for related categories of spaces as well as discussing preservation of open mappings by various topological constructions.
Reviewer: K.I.Rosenthal

MSC:

18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
54B30 Categorical methods in general topology
18B30 Categories of topological spaces and continuous mappings (MSC2010)
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)

Citations:

Zbl 0537.18001
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References:

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