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On the Relationship between Filter Spaces and Equilogical Spaces
, 1998
"... It was already known that the category of T 0 topological spaces is not itself cartesian closed, but can be embedded into the cartesian closed categories FIL of filter spaces and EQU of equilogical spaces where the latter embeds into the cartesian closed category ASSM of assemblies over algebraic la ..."
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It was already known that the category of T 0 topological spaces is not itself cartesian closed, but can be embedded into the cartesian closed categories FIL of filter spaces and EQU of equilogical spaces where the latter embeds into the cartesian closed category ASSM of assemblies over algebraic lattices. Here, we first clarify the notion of filter spacethere are at least three versions FIL a ' FIL b ' FIL c in the literature. We establish adjunctions between FIL a and ASSM and between FIL c and ASSM, and show that FIL b and FIL c are equivalent to reflective full subcategories of ASSM. The corresponding categories FIL b 0 and FIL c 0 of T 0 spaces are equivalent to full subcategories of EQU. Keywords: Categorical models and logics, domain theory and applications Author's address: Reinhold Heckmann, FB 14  Informatik, Universitat des Saarlandes, Postfach 151150, D66041 Saarbrucken, Germany Phone: +49 681 302 2454 Fax: +49 681 302 3065 email: heckmann@cs.un...
JOYAL’S ARITHMETIC UNIVERSE AS LISTARITHMETIC PRETOPOS
"... Abstract. We explain in detail why the notion of listarithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel’s incompleteness results. We motivate this definition for three ..."
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Abstract. We explain in detail why the notion of listarithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel’s incompleteness results. We motivate this definition for three reasons: first, Joyal’s arithmetic universes are listarithmetic pretopoi; second, the initial arithmetic universe among Joyal’s constructions is equivalent to the initial listarithmetic pretopos; third, any listarithmetic pretopos enjoys the existence of free internal categories and diagrams as required to prove Gödel’s incompleteness. In doing our proofs we make an extensive use of the internal type theory of the categorical structures involved in Joyal’s constructions. The definition of listarithmetic pretopos is equivalent to the general one that I came to know in a recent talk by André Joyal. 1.
Separation Classes of Equilogical Spaces
 Workshop on Domains IV (Rolandseck
, 1999
"... The category TOP of topological spaces is not cartesian closed, but can be embedded into the cartesian closed category ASSM of assemblies over algebraic lattices, which is a generalisation of Scott's category EQU of equilogical spaces. In this paper, we identify cartesian closed subcategorie ..."
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The category TOP of topological spaces is not cartesian closed, but can be embedded into the cartesian closed category ASSM of assemblies over algebraic lattices, which is a generalisation of Scott's category EQU of equilogical spaces. In this paper, we identify cartesian closed subcategories of assemblies which correspond to wellknown separation properties of topology: T 0 , T 1 , Hausdorff, completely Hausdorff, totally disconnected, completely regular, zerodimensional. 1 Introduction It is wellknown that the categories TOP of topological spaces and TOP 0 of T 0 topological spaces are not cartesian closed, i.e., do not admit a function space construction such that typed calculus could be interpreted in the category. In December 1996, Dana Scott proposed to remedy the situation by extending TOP 0 to the cartesian closed category EQU of equilogical spaces [12, 1, 4]. There are several equivalent descriptions of EQU; one of them is by modest sets over algebraic lattices [1]....
Ordered Equilogical Spaces and Equilogical DSpaces
, 1998
"... Equilogical spaces form a cartesian closed complete category that contains all T_0 spaces. We first present the cartesian closed complete subcategory of ordered equilogical spaces that still contains all T_0 spaces, but excludes certain pathological equilogical spaces. Then, we further restrict to t ..."
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Equilogical spaces form a cartesian closed complete category that contains all T_0 spaces. We first present the cartesian closed complete subcategory of ordered equilogical spaces that still contains all T_0 spaces, but excludes certain pathological equilogical spaces. Then, we further restrict to the cartesian closed complete subcategory of equilogical dspaces that still contains all dcpo's and all T_1 spaces. Its main advantage is that equilogical dspaces with least element admit a least fixed point operator.
Injective Convergence Spaces and Equilogical Spaces via Pretopological Spaces
 MFPS 2005
, 2005
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