On injectivity in locally presentable categories

Adámek, Jiří; Rosický, Jiří

doi:10.1090/S0002-9947-1993-1085935-2

On injectivity in locally presentable categories
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by Jiří Adámek and Jiří Rosický

Trans. Amer. Math. Soc. 336 (1993), 785-804

DOI: https://doi.org/10.1090/S0002-9947-1993-1085935-2

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Abstract:

Classes of objects injective w.r.t. specified morphisms are known to be closed under products and retracts. We prove the converse: a class of objects in a locally presentable category is an injectivity class iff it is closed under products and retracts. This result requires a certain large-cardinal principle. We characterize classes of objects injective w.r.t. a small collection of morphisms: they are precisely the accessible subcategories closed under products and $\kappa$-filtered colimits. Assuming the (large-cardinal) Vopênka’s principle, the accessibility can be left out. As a corollary, we solve a problem of ${\text {L}}$. Fuchs concerning injectivity classes of abelian groups. Finally, we introduce a weak concept of reflectivity, called cone reflectivity, and we prove that under Vopênka’s principle all subcategories of locally presentable categories are cone reflective. Several open questions are formulated, e.g., does each topological space have a largest (non-${T_2}$) compactification?

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