Representability theorems via metric techniques
Kabeer Manali Rahul
TL;DR
This work develops a metric-based approach to Brown representability in triangulated categories by introducing and exploiting (weak) $\mathcal{G}$-approximability, a generalisation of Neeman’s approximability using a generating sequence $\mathcal{G}$ and an orthogonal metric. The main idea is to study the closure of the compacts $\overline{\mathsf{T}^c}$ and its bounded part $\mathsf{T}^b_c$ via restricted Yoneda functors, establishing full faithfulness and explicit descriptions of the essential images under suitable hypotheses. The paper then applies these representability theorems to algebraic-geometry contexts, obtaining new results for homological and cohomological functors on $\mathbf{D}^{b}_{\operatorname{coh}}(X)$ and related categories such as $\mathbf{D}_{\operatorname{Qcoh}}(X)$, $\mathbf{D}^{\operatorname{perf}}(X)$, and homotopy categories like $\mathbf{K}(\operatorname{Inj}-X)$ and $\mathbf{K}_{m}(\operatorname{Proj}-X)$. Through concrete examples (e.g. BG$_m$ and stacks) and noncommutative analogues, the results demonstrate the breadth of representability phenomena beyond categories with a single compact generator. Overall, the framework provides a robust tool for identifying representable functors and their representing objects in diverse geometric and noncommutative settings, with implications for derived categories in algebraic geometry and beyond.
Abstract
We prove new Brown representability theorems for triangulated categories using metric techniques as introduced in the work of Neeman. In the setting of algebraic geometry, this gives us new representability theorems for homological and cohomological functors on the bounded derived category of coherent sheaves. To prove this result, we introduce a generalisation of the notion of an approximable triangulated category.