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

DOI:
https://doi.org/10.1090/S0002-9947-1984-0742429-3

MathSciNet review:
742429

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9947-1984-0742429-3

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