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Topological space objects in a topos II: ɛ-Completeness and ɛ-cocompleteness

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Abstract

It is well known that topoi satisfy strong internal completeness and cocompleteness conditions: Lawvere [4] announced the existence of internal Kan extensions; proofs may be found in Kock and Wraith [3] and Diaconescu [2]. In this paper I give an explicit construction of the limit of an internal functor and lift the completeness and cocompleteness of ɛ to the category of topological space objects in ɛ defined by internalizing the definition in terms of open sets (as in [7] and [8]).

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References

  1. BENABOU, JEAN: Introduction to Bicategories, in Reports of the Midwest Category Theory Seminar, Lecture Notes in Mathematics 47. Berlin, Heidelberg, and New York: Springer 1967.

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  2. DIACONESCU, RADU: Change of Base for Some Toposes, Thesis, Dalhousie 1973.

  3. KOCK, ANDERS and WRAITH, GAVIN: Elementary Toposes, Lecture Notes Series 30, Aarhus Universitat Matematisk Institut 1970.

  4. LAWVERE, F. WILLIAM: Quantifiers and Sheaves, Actes du Congrès Int. des Math. Nice 1970, I, 329–334.

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  5. OSIUS, GERHARD: Categorical Set Theory: a Characterization of the Category of Sets, J. Pure and Appl.Alg. Vol.4, 79–120 (1974).

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  6. OSIUS, GERHARD: The Internal and External Aspect of Logic and Set Theory in Elementary Topoi, preprint, 1974.

  7. STOUT, LAWRENCE: General Topology in an Elementary Topos, Thesis, University of Illinois, 1974.

  8. STOUT, LAWRENCE: Topological Space Objects in a Topos I: Variable Spaces for Variable Sets. To appear.

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Authors and Affiliations

  1. Department of Mathematics, McGill University, Station A, Box 6070, H3C 3G1, Montréal, Québec, Canada

    Lawrence Neff Stout

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  1. Lawrence Neff Stout
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Parts of this work appeared in the author's thesis at the University of Illinois, 1974.

Research partially supported by a grant from the Ministry of Education of the Province of Québec.

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Stout, L.N. Topological space objects in a topos II: ɛ-Completeness and ɛ-cocompleteness. Manuscripta Math 17, 1–14 (1975). https://doi.org/10.1007/BF01154279

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  • DOI: https://doi.org/10.1007/BF01154279

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