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- W163404 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 $$ Botimes vec{A} $$ 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, $$ Botimes vec{A},=,{lim}_rightarrow pin H{A_p}. $$ In case X is finite and $$ v(x)=P, downarrow,f(x)({rm where} f(x)in P)$$ for each $$xin X$$ , then $$ Botimes vec{A}=prod(A_{f_(x)}mid {xin X}) $$ . 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 $$ B={lim_rightarrow iin I}{B_i}$$ where all the $$ {B_i}$$ are finitely presented (cf. Proposition 2.4.6) and then $$ Botimes vec{A} $$ is defined as the corresponding directed colimit of the $$ {B_i}otimes vec{A} $$ . That this can be done, and that the resulting functor $$ Bmapsto Botimes vec{A}$$ preserves all small directed colimits, will follow from Proposition 1.4.2." @default.
- W163404 created "2016-06-24" @default.
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- W163404 date "2011-01-01" @default.
- W163404 modified "2023-09-25" @default.
- W163404 title "Boolean Algebras That Are Scaled with Respect to a Poset" @default.
- W163404 doi "https://doi.org/10.1007/978-3-642-21774-6_2" @default.
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