Categories of Partial Frames
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چکیده
This article discusses the basic categorical algebra for categories of partial frames. Categories of partial frames are labelled by subset selectors that indicate which joins exist. Constructions for limits, colimits, and free functors connecting various categories of partial frames are given. Examples of partial frame categories are given. Subset selectors which preserve surjections are virtually the same as rules which select all subsets smaller than a given cardinal. Preface The article’s goal is to describe categories of partial frames. A partial frame is a meetsemilattice in which certain distinguished joins exist and finite meets distribute over distinguished joins. This is made precise in Subsection 2.1. Particular types of “partial frames” have already appeared in frame theoretic literature. Madden [19] and Madden & Molitor [20] use κ-frames for any regular cardinal κ to draw useful frame-theoretic conclusions: many monoreflections on the category of Tychonoff locales are produced, and epimorphisms of frames (monomorphisms of locales) are identified. Johnstone and Vickers [11] and Banaschewski [1] and [4] use preframes – meet semilattices with directed joins and a distributive law – to simplify calculation of colimits of frames and produce “the shortest known proof of the Tychonoff Theorem.” The article of Paseka [24] has similar goals to this article, so a key difference deserves mention. Paseka only considers subset selectors F which have the feature that if f : A → B is a surjective meetsemilattice homomorphism, FB = {f(S) : S ∈ FA}. Paseka gives no examples other than cardinalities which satisfy this condition, and Propositions 37 and 39 (proved below) make me doubt that there are other examples. If this is the case, [24] does not prove anything that was not already in [19]. In contrast, this article’s definition of subset selector (Definition 26) is more general. The last section of the paper describes several interesting classes of subset selectors. Moreover, the machinery developed in this article does not depend on the precise definition of meetsemilattice. For example it does not depend on whether one requires meetsemilattices to have a top, bottom or any combination thereof. In fact, most of the important results still hold if one replaces meetsemilattices with Date: March 16, 2005. 1991 Mathematics Subject Classification: 06d22, 06b23, 18c15, 18c20, 18a20, 18a30.
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تاریخ انتشار 2005