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From intuitionistic to pointfree topology: on the foundation of homotopy theory
, 2005
"... Brouwer’s pioneering results in topology, e.g. invariance of dimension, were developed within a classical framework of mathematics. Some years later he explained ..."
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Brouwer’s pioneering results in topology, e.g. invariance of dimension, were developed within a classical framework of mathematics. Some years later he explained
Minimal Invariant Spaces in Formal Topology
 The Journal of Symbolic Logic
, 1996
"... this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a pointfree formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a ..."
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this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a pointfree formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a suitable formal topology, and the "existence" of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is "similar in structure" to the topological proof [6, 8], but which uses a simple algebraic remark (proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements. 1 Construction of Minimal Invariant Subspace
Compact Spaces and Distributive Lattices
 Journal of Pure and Applied Algebra
"... Introduction This note presents a general construction (theorem 2.1) connecting compact locales and distributive lattices. This allows us to reduce results about compactness of locales to theorems about distributive lattices. We give two applications: a reduction of the localic version of Tychono's ..."
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Introduction This note presents a general construction (theorem 2.1) connecting compact locales and distributive lattices. This allows us to reduce results about compactness of locales to theorems about distributive lattices. We give two applications: a reduction of the localic version of Tychono's theorem [7], that states that a product of compact locales is locales, to a result about coproduct of distributive lattices, and a proof of the localic version of Steenrod's theorem, that states that an inverse limits of compact locales is compact. Our proof of Tychono's theorem does not use any decidability hypotheses on the index set . The localic version of Steenrod's theorem appears in [6], but with a proof that uses classical logic. One noteworthy feature of our arguments is that they can be formulated both in topos theory and in a predicative theory such as CZF [1, 2]. Another method to analyse compactness is provided by the theory of preframes [3, 9, 11], that are structures cl
Galois Groupoids and Covering Morphisms in Topos Theory
"... The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given ..."
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Cited by 2 (2 self)
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The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given by a pair of adjoint functors, and (2) to discuss a good notion of covering morphism of a topos E over S which is general enough to include, in addition to the covering projections determined by the locally constant objects, also the unramified morphisms of topos theory given by those local homeomorphisms which are at the same time complete spreads in the sense of BungeFunk (1996, 1998).
Notions Invariant By Change Of Bases
, 2001
"... One importance of theory of sheaves for constructive mathematics is that it provides a general expression of the algebraic method of "adding new indeterminates". The method of "change of base", or moving in a suitable topos of sheaves over a formal space, can indeed be interpreted as the addition o ..."
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One importance of theory of sheaves for constructive mathematics is that it provides a general expression of the algebraic method of "adding new indeterminates". The method of "change of base", or moving in a suitable topos of sheaves over a formal space, can indeed be interpreted as the addition of new unknown quantities submitted to some given constraints. This method compensates the absence of choice (an example is provided in [6]), and gives a constructive way to add ideal elements such as prime ideals. This method requires some care however: adding these quantities may add new sequences for instance, and some facts that did hold before the addition of these quantities may not hold anymore after. It is thus quite important to recognize the concepts that are invariant under such an operation. This motivates for instance the notion of "geometrical statement": if a concept is expressed by a geometrical statement, it is invariant. The purpose of this note is to a