Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Localization and sheaf reflectors


Authors: J. Lambek and B. A. Rattray
Journal: Trans. Amer. Math. Soc. 210 (1975), 279-293
MSC: Primary 18C15
MathSciNet review: 0447364
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Abstract: Given a triple $ (S,\eta ,\mu )$ on a category $ \mathcal{A}$ with equalizers, one can form a new triple whose functor $ Q$ is the equalizer of $ \eta S$ and $ S\eta $. Fakir has studied conditions for $ Q$ to be idempotent, that is, to determine a reflective subcategory of $ \mathcal{A}$. Here we regard $ S$ 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 $ \mathcal{A}$ from an injective object $ I$, taking $ S = {I^{{I^{( - )}}}}$. We show that this reflector preserves finite limits and that the sheaf reflector for a topology in $ \mathcal{A}$ can be obtained in this way. We also show that sheaf reflectors in functor categories can be obtained from a triple of the form $ S = {I^{( - ,I)}},I$ 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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DOI: http://dx.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