Several authors have studied the filtered colimit closure \documentclass[12pt]{minimal}
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\begin{document}$\varinjlim\mathcal{B}$\end{document} of a class \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{B}$\end{document} of finitely presented modules. Lenzing called \documentclass[12pt]{minimal}
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\begin{document}$\varinjlim\mathcal{B}$\end{document} the category of modules with support in \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{B}$\end{document}, and proved that it is equivalent to the category of flat objects in the functor category \documentclass[12pt]{minimal}
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\begin{document}$(\mathcal{B}^\mathrm{op},\mathsf{Ab})$\end{document}. In this paper, we study the category \documentclass[12pt]{minimal}
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\begin{document}$({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}$\end{document} of modules with cosupport in \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{B}$\end{document}. We show that \documentclass[12pt]{minimal}
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\begin{document}$({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}$\end{document} is equivalent to the category of injective objects in \documentclass[12pt]{minimal}
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\begin{document}$(\mathcal{B},\mathsf{Ab})$\end{document}, and thus recover a classical result by Jensen-Lenzing on pure injective modules. Works of Angeleri-Hügel, Enochs, Krause, Rada, and Saorín make it easy to discuss covering and enveloping properties of \documentclass[12pt]{minimal}
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\begin{document}$({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}$\end{document}, and furthermore we compare the naturally associated notions of \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{B}$\end{document}-coherence and \documentclass[12pt]{minimal}
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\begin{document}$\mathcal{B}$\end{document}-noetherianness. Finally, we prove a number of stability results for \documentclass[12pt]{minimal}
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\begin{document}$\varinjlim\mathcal{B}$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}$\end{document}. Our applications include a generalization of a result by Gruson-Jensen and Enochs on pure injective envelopes of flat modules.
