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
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.
DIACONESCU, RADU: Change of Base for Some Toposes, Thesis, Dalhousie 1973.
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OSIUS, GERHARD: The Internal and External Aspect of Logic and Set Theory in Elementary Topoi, preprint, 1974.
STOUT, LAWRENCE: General Topology in an Elementary Topos, Thesis, University of Illinois, 1974.
STOUT, LAWRENCE: Topological Space Objects in a Topos I: Variable Spaces for Variable Sets. To appear.
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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