Brown representability test problems in locally Grothendieck categories (Q421478)

Let \(\mathcal{K}\) be a \(\kappa\)-compactly generated triangulated category with \(\mathcal{K}_\kappa\) its representative small full subcategory of \(\kappa\)-compact objects. Further, let \(E_\mathcal{K}: \mathcal{K}\to \mathcal{A}b^{\mathcal{K}_\kappa^{\mathrm{op}}}\) be the Yoneda-like functor sending \(K\) to \(\mathcal{K}(-,K)\), but restricted to \(\mathcal{K}_\kappa\). The \(\kappa\)-Brown representability question is whether or not \(E_\mathcal{K}\) is full, and this is central to questions concerning the representability if cohomology theories in this context. Rosický and Muro have formulated two test problems for Grothendieck categories for which a positive solution would imply a positive answer to the Brown representability question in those contexts. In this paper, it is shown that these test problems have a negative solution in the locally Grothendieck category, \(\bigcup_{\kappa\geq \aleph_0}{\mathcal Mod}-K^\kappa\), where \(K\) is a field.