Higher weak (co)limits, adjoint functor theorems, and higher Brown representability

  • Hoang Kim Nguyen

    Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
  • George Raptis

    Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
  • Christoph Schrade

    Mathematisches Institut, WWU Münster, 48149 Münster, Germany
Higher weak (co)limits, adjoint functor theorems, and higher Brown representability cover
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Abstract

We prove general adjoint functor theorems for weakly (co)complete nn-categories. This class of nn-categories includes the homotopy nn-categories of (co)complete \infty-categories, so these nn-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) nn-categories and prove a Brown representability theorem for localizations of compactly generated nn-categories. This class of nn-categories includes the homotopy nn-categories of presentable \infty-categories if n2n \geq 2, and the homotopy nn-categories of presentable stable \infty-categories for any n1n \geq 1.

Cite this article

Hoang Kim Nguyen, George Raptis, Christoph Schrade, Higher weak (co)limits, adjoint functor theorems, and higher Brown representability. Doc. Math. 27 (2022), pp. 1369–1420

DOI 10.4171/DM/900