We prove general adjoint functor theorems for weakly (co)complete \(n\)-categories. This class of \(n\)-categories includes the homotopy \(n\)-categories of (co)complete \(\infty\)-categories, so these \(n\)-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) \(n\)-categories and prove a Brown representability theorem for localizations of compactly generated \(n\)-categories. This class of \(n\)-categories includes the homotopy \(n\)-categories of presentable \(\infty\)-categories if \(n \geq 2\), and the homotopy \(n\)-categories of presentable stable \(\infty \)-categories for any \(n \geq 1\).

18N60, 55U35, 18G80, 55P99

adjoint functor theorem, Brown representability, higher category

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Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Mathematisches Institut, WWU Münster, 48149 Münster, Germany