Localization and sheaf reflectors

Authors:
J. Lambek and B. A. Rattray

Journal:
Trans. Amer. Math. Soc. **210** (1975), 279-293

MSC:
Primary 18C15

DOI:
https://doi.org/10.1090/S0002-9947-1975-0447364-0

MathSciNet review:
0447364

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Abstract: Given a triple on a category with equalizers, one can form a new triple whose functor is the equalizer of and . Fakir has studied conditions for to be idempotent, that is, to determine a reflective subcategory of . Here we regard as the composition of an adjoint pair of functors and give several new such conditions. As one application we construct a reflector in an elementary topos from an injective object , taking . We show that this reflector preserves finite limits and that the sheaf reflector for a topology in can be obtained in this way. We also show that sheaf reflectors in functor categories can be obtained from a triple of the form injective, which we studied in a previous paper. We deduce that the opposite of a sheaf subcategory of a functor category is tripleable over Sets.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0447364-0

Keywords:
Localization,
triple,
reflective subcategory,
injective,
sheaf,
cartesian closed category,
topos

Article copyright:
© Copyright 1975
American Mathematical Society