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Limits and colimits of convexity spaces. (English) Zbl 0637.52001

The author studies the categories of preconvexity spaces with Darboux maps and prealigned spaces with monotone maps, defined as follows: a preconvexity space is a set together with a collection of subsets (called convex sets) closed under intersection of nonempty subfamilies, a function between two preconvexity spaces is said to be a Darboux (resp. monotone) map if it preserves (resp. reflects) convex sets; when monotone maps are considered, preconvexity spaces are called, instead, prealigned. Thus, a convex prealigned space is an aligned space in the sense of R. E. Jamison-Waldner [Lect. Notes Pure Appl. Math. 76, 113-150 (1982; Zbl 0482.52001)]. If the family of convex sets is also closed under nested unions, the resulting structure is called a convexity (or an aligned) space. All these notions are illustrated by means of examples pertaining to several branches of mathematics. The limits and colimits in these categories and some of their relevant subcategories are extensively studied. In particular, it is proved that any preconvexity space has a unique representation as a coproduct of connected preconvexity spaces (“connected” means that for any two distinct points x, y, there is a chain of convex sets \(A_ 0,A_ 1,...,A_ n\) such that \(A_ i\cap A_{i+1}\neq \emptyset\), \(x\in A_ 0\), \(y\in A_ n)\). Finally, the author mentions that a homology theory of preconvexity spaces can be based on the notion of downclosed convexity space (regarded as a generalization of simplicial complex), which is a convexity space with the additional property that every subset of a convex set is also convex.
Reviewer: J.E.Martinez-Legaz

MSC:

52A01 Axiomatic and generalized convexity
18B99 Special categories
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)

Citations:

Zbl 0482.52001
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References:

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