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Notions of topos

  • Ross Street (a1)
Abstract

A Grothendieck topos has the property that its Yoneda embedding has a left-exact left adjoint. A category with the latter property is called lex-total. It is proved here that every lex-total category is equivalent to its category of canonical sheaves. An unpublished proof due to Peter Freyd is extended slightly to yield that a lex-total category, which has a set of objects of cardinality at most that of the universe such that each object in the category is a quotient of an object from that set, is necessarily a Grothendieck topos.

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References
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[1]Artin, M., Grothendieck topologies (Department of Mathematics, Harvard University, Massachusetts, 1962).
[2]Freyd, P.J. and Kelly, G.M., “Categories of continuous functors, I”, J. Pure Appl. Algebra 2 (1972), 169191.
[3]Grothendieck, A. et Verdier, J.L., “Topos”, Théorie des topos et cohomologie etale des schemas, Expose IV, 299519 (Séminaire de Géométrie Algébrique du Bois Marie, 1963/64. Lecture Notes in Mathematics, 269. Springer-Verlag, Berlin, Heidelberg, New York, 1972).
[4]Johnstone, P.T., Topos theory (London Mathematical Society Monographs, 10. Academic Press, London, New York, San Francisco, 1977).
[5]Street, Ross, “Cosmoi of internal categories”, Trans. Amer. Math. Soc. 258 (1980), 271318.
[6]Street, Ross, “Extract from a letter on legitimacy of categories of presheaves”, Diagrammes (to appear).
[7]Street, Ross and Walters, Robert, “Yoneda structures on 2-categories”, J. Algebra 50 (1978), 350379.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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