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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Localization at injectives in complete categories

Authors: J. Lambek and B. A. Rattray
Journal: Proc. Amer. Math. Soc. 41 (1973), 1-9
MSC: Primary 18A35
MathSciNet review: 0414651
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Abstract: We consider a complete category $ \mathcal{A}$. For each object $ I$ of $ \mathcal{A}$ we define a functor $ Q:\mathcal{A} \to \mathcal{A}$ and obtain a necessary and sufficient condition on $ I$ for $ Q$, after restricting its codomain, to become a reflector of $ \mathcal{A}$ onto the limit closure of $ I$. In particular, this condition is satisfied if $ I$ is injective in $ \mathcal{A}$ with regard to equalizers. Among the special cases of such reflectors are: the reflector onto torsion-free divisible objects associated to an injective $ I$ in $ \operatorname{Mod} R$; the Samuel compactification of a uniform space; the Stone-Čech compactification.

We give a second description of $ Q$ in terms of a triple on sets. If $ I$ is injective and the functor $ Q$ is equivalent to the identity then, under a few extra conditions on $ \mathcal{A},{\mathcal{A}^{{\text{op}}}}$ is triplable over sets with regard to the functor taking $ A$ to $ \mathcal{A}(A,I)$.

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Keywords: Localization, reflective subcategory, injective, triple, torsion theory
Article copyright: © Copyright 1973 American Mathematical Society