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Boolean Algebras That Are Scaled with Respect to a Poset

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2029)

Abstract

Our main result, CLL (Lemma 3.4.2), involves a construction that turns a diagram \(\vec{A} \) indexed by a poset P, from a category A, to an object of A, called a condensate \(\vec{A} \) (cf. Definition 3.1.5). A condensate of \(\vec{A} \) will be written in the from \( B\otimes \vec{A} \) where B is a Boolean algebra with additional structure –we shall say a P-scaled Boolean algebra (Definition 2.2.3). It will turn out (cf. Proposition 2.2.9) that P-scaled Boolean algebra are the dual objects of topological objects called P-normed Boolean algebra (cf.Definition 2.2.1). By definition, a P-normed topological space is a topological space X endowed with a map (the “norm function”) from X to Id P which is continuous with respect to the given topology of X and the Scott topology on Id P is a one-point space with norm an ideal H of P and B is the corresponding P-scaled Boolean algebra, \( B\otimes \vec{A}\,=\,{lim}_\rightarrow p\in H{A_p}. \) In case X is finite and \( v(x)=P\, \downarrow\,f(x)({\rm where} f(x)\in P)\) for each \(x\in X\), then \( B\otimes \vec{A}=\prod(A_{f_(x)}\mid {x\in X}) \). The latter situation describes the case where B is a finitely presented P-scaled Boolean algebra (cf. Definition 2.4.1 and Corollary 2.4.7). In the general case, there is a directed colimit representation \( B={lim_\rightarrow i\in I}{B_i}\) where all the \( {B_i}\) are finitely presented (cf. Proposition 2.4.6) and then \( B\otimes \vec{A} \) is defined as the corresponding directed colimit of the \( {B_i}\otimes \vec{A} \). That this can be done, and that the resulting functor \( B\mapsto B\otimes \vec{A}\) preserves all small directed colimits, will follow from Proposition 1.4.2.

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Correspondence to Pierre Gillibert .

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© 2011 Springer-Verlag Berlin Heidelberg

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Gillibert, P., Wehrung, F. (2011). Boolean Algebras That Are Scaled with Respect to a Poset. In: From Objects to Diagrams for Ranges of Functors. Lecture Notes in Mathematics(), vol 2029. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21774-6_2

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