F.W. Lawvere, Categories of space and quantity, in: J. Echeverria et al., editors, The Space of Mathematics, de Gruyter, Berlin, New York, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....LTop S of Top S consisting of the locally connected toposes and show these 2 categories to be extensive. This is of independent interest, as it shows that Top S and LTop S are indeed suitable domains for considering intensive and extensive quantities on toposes, as suggested by Lawvere [27, 28]. An intensive quantity on an extensive 2 category K is a pseudofunctor C : K CAT which preserves finite products. We then consider three particular examples of intensive quantities on toposes corresponding respectively to local homeomorphisms, covering projections, and unramified morphisms; ....

F.W. Lawvere, Categories of space and quantity, in: J. Echeverria et al., editors, The Space of Mathematics, de Gruyter, Berlin, New York, 1992.

....In the first, we derive the Jibladze Johnhnstone Theorem (characterizing relatively closed sublocales) by analyzing a certain pair of adjoint functors. We use this, in the second section, to derive a relationship between certain intensive and extensive quantities (in the sense of Lawvere [10]) on an open locale; the extensive quantities in question being certain frame distributions , suggested by Lawvere [9] 10] and studied by Bunge and Funk in [1] We give an alternative proof of their result: identifying these frame distributions on a locale M with certain sublocales of M . A ....

....certain pair of adjoint functors. We use this, in the second section, to derive a relationship between certain intensive and extensive quantities (in the sense of Lawvere [10] on an open locale; the extensive quantities in question being certain frame distributions , suggested by Lawvere [9] [10], and studied by Bunge and Funk in [1] We give an alternative proof of their result: identifying these frame distributions on a locale M with certain sublocales of M . A preliminary version Frame distributions and support was made available on the internet al..ready in January 1996 (as ....

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F.W. Lawvere, Categories of space and quantity. In J. Echeverr'ia, A. Ibarra, and T. Mormann (eds.). The Space of Mathematics. Berlin: de Gruyter

....that is almost equivalent to Fox s notion of a complete spread. Fox had introduced complete spreads as a framework by which to study topologically ramified covers. It may interest the reader that a cosheaf can be equivalently regarded as a topos distribution in the sense of Lawvere [20, 21]. We shall not pursue this connection with topos distributions, but the reader can consult [6] and references cited therein) for information and recent Received by the editors 2000 December 31 and, in revised form, 2001 December 17. Published on 2001 December 21 in the volume of articles from ....

F. W. Lawvere. Categories of space and of quantity. In J. Echeverria et al., editors, The Space of Mathematics, pages 14--30. W. de Gruyter, Berlin-New York, 1992.

....0. INTRODUCTION By an S valued distribution on a topos E bounded over a base topos S it is meant here a cocontinuous S indexed functor : E S. Since introduced by F. W. Lawvere in 1983, considerable progress has been made in the study of distributions on toposes from a variety of viewpoints [19, 15, 24, 5, 6, 12, 7, 8, 9]. However, much work still remains to be done in this area. The purpose of this paper is to deepen our understanding of topos distributions by exploring a (dual) lattice theoretic notion of distribution algebra. We characterize the distribution algebras in E relative to S as the S bicomplete ....

F. W. Lawvere, Categories of space and quantity, in : J. Echeverr  ia et al. eds. The Space of Mathematics, W. de Gruyter, Berlin (1992) 14-30.

....In particular, the theory of complete spreads in topos theory ought to provide a basis for an investigation of branched covers in topos theory, which is the purpose of this paper. We shall adopt Lawvere s proposal and refer to a cocontinuous functor from one topos into another as a distribution [15, 16]. The canonical correspondence of distributions with complete spreads [4] has a direct influence on our investigation of branched covers. Indeed, our study of branched covers will involve a certain subtopos of a given topos that has an interesting interpretation in terms of distributions; we shall ....

F. W. Lawvere. Categories of space and of quantity. In J. Echeverria et al., editors, The Space of Mathematics, pages 14--30. W. de Gruyter, Berlin-New York, 1992.

....00 m ( f f 0 ) G f 00 and D GC G(C 0 (C 00 ) D 0 ( D 00 D 0 ( GC 00 f G ( f 0 (GC 00 ) q D 0 ( f 00 and D GC X G(C 0 (C 00 ) f G p 2 ut This seems to be a folklore Lawvere has stated this result in his lectures in 1990, c.f. [31]. Casley et al. 13] describe this too. Also see Ambler s thesis [4] for a related observation. In fact, a more abstract point of view is available, in terms of fibrations. Hermida [24] has shown that, if we have a fibred ccc p : E B , B with finite products and p with Cons B products, then so ....

Lawvere, F.W. (1992), Categories of space and of quantity, in "The Space of Mathematics: Philosophical, Epistemological and Historical Explorations", pp. 14--30, DeGruyter, Berlin.

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