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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The family approach to total cocompleteness and toposes


Author: Ross Street
Journal: Trans. Amer. Math. Soc. 284 (1984), 355-369
MSC: Primary 18B25; Secondary 18A30, 18A32, 18A35, 18F10, 18F20
MathSciNet review: 742429
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Abstract: A category with small homsets is called total when its Yoneda embedding has a left adjoint; when the left adjoint preserves pullbacks, the category is called lex total. Total categories are characterized in this paper in terms of special limits and colimits which exist therein, and lex-total categories are distinguished as those which satisfy further exactness conditions. The limits involved are finite limits and intersections of all families of subobjects. The colimits are quotients of certain relations (called congruences) on families of objects (not just single objects). Just as an arrow leads to an equivalence relation on its source, a family of arrows into a given object leads to a congruence on the family of sources; in the lex-total case all congruences arise in this way and their quotients are stable under pullback. The connection with toposes is examined.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0742429-3
PII: S 0002-9947(1984)0742429-3
Keywords: Total and lex-total category, exact category, factorization of families, Grothendieck topos, finitely presentable, universal extremal epimorphic family
Article copyright: © Copyright 1984 American Mathematical Society