Carles Casacuberta, Amnon Neeman
We exhibit a triangulated category T having both products and coproducts, and a triangulated subcategory S of T which is both localizing and colocalizing, for which neither a Bousfield localization nor a colocalization exists. It follows that neither the category S nor its dual satisfy Brown representability. Our example involves an abelian category whose derived category does not have small Hom-sets.
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We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion into K(Ab) does not have a right adjoint.
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 $\in…
We show that the dual of the homotopy category of projective modules over an arbitrary ring satisfies Brown representability.
Brown representability for directed graphs
该研究证明满足可加性与Mayer-Vietoris公理的、从有限有向图同伦范畴到阿贝尔群的反变函子具有可表示性。
These exercises complement my notes "Derived categories, resolutions, and Brown representability".
For a triangulated category with products we develop a method for constructing a nice set of cogenerators, allowing us to prove a formal criterion in order to satisfy Brown representability for covariant functors. We apply this criterion for showing that both homotopy category of projective modules and homotopy category of projective objects in a category of functors from a small category to a module category satisfy this kind of representability. In particular, homotopy category of projective complexes satisfies Brown representability for covariant functors.
We call product generator of an additive category a fixed object satisfying the property that every other object is a direct factor of a product of copies of it. In this paper we start with an additive category with products and images, e.g. a module category, and we are concerned with the homotopy category of complexes with entries in that additive category. We prove that Brown representability theorem is valid for the dual of the homotopy category if and only if the initial additive category has a product generator.
Definable functors and Brown--Adams representability
本文研究了环的导出范畴中布朗-阿德斯可代表性的条件，通过可定义函子的同构转换，证明了纯全局维数的关系，并给出了茶杯猜想的新证明及协调环的生成假设新结论。
Triangulated categories with a single compact generator and two Brown representability theorems
该研究构建可逼近三角范畴的一般形式体系，证明两个相关可表示性定理，为三角范畴领域的可表示性研究提供理论支撑。
In this paper we give necessary and sufficient conditions for a functor to be representable in a strongly generated triangulated category which has a linear action by a graded ring, and we discuss some applications and examples.
These notes are based on a series of five lectures given during the summer school ``Interactions between Homotopy Theory and Algebra'' held at the University of Chicago in 2004.
Brown--Adams representability for triangulated categories with locally coherent cohomology
本文针对具有局部凝聚上同调的三角范畴，证明了Brown-Adams可表示性定理的变体，给出了函子由对象表示的条件。
We show that the homotopy category of a combinatorial stable model category $\ck$ is well generated. It means that each object $K$ of $\Ho(\ck)$ is an iterated weak colimit of $λ$-compact objects for some cardinal $λ$. A natural question is whether each $K$ is a weak colimit of $λ$-compact objects. We show that this is related to (generalized) Brown representability of $\Ho(\mathcal K)$.
Consider a complete abelian category which has an injective cogenerator. If its derived category is left--complete we show that the dual of this derived category satisfies Brown representability. In particular this is true for the derived category of an abelian AB$4^*$-$n$ category, for the derived category of quasi--coherent sheaves over a nice enough scheme (including the projective finitely dimensional space) and for the full subcategory of derived category of all sheaves over an algebraic stack consisting from complexes with quasi--coherent cohomology.
In a well generated triangulated category T, given a regular cardinal a, we consider the following problems: given a functor from the category of a-compact objects to abelian groups that preserves products of <a objects and takes exact triangles to exact sequences, is it the restriction of a representable functor in T? Is every natural transformation between two such restricted representable functors induced by a map between the representatives? If the answer to both questions is positive we say that T satisfies a-Adams representability. A classical result going back to Brown and Adams shows t…
Representability theorems via metric techniques
该研究利用Neeman的度量技术证明三角范畴的新布朗可表性定理，在代数几何中为凝聚层有界导出范畴的同调与上同调函子提供新可表性定理，并推广了可逼近三角范畴概念。
We define quasi--locally presentable categories as big unions of coreflective subcategories which are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi--locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi--locally presentable and we obtain a new proof of Brown representability theorem. Examples of functors which are not representable are also given.
Given a fixed tensor triangulated category S we consider triangulated categories T together with an S-enrichment which is compatible with the triangulated structure of T. It is shown that, in this setting, an enriched analogue of Brown representability holds when both S and T are compactly generated. A natural class of examples of such enriched triangulated categories categories are module categories over separable monoids in S. In this context we prove a version of the Eilenberg-Watts theorem for exact coproduct and copower preserving S-functors, i.e., we show that any such functor between th…
Torsion and complete dualizable objects in tensor-triangulated categories over a Noetherian ring
本文研究了在Noetherian环作用下张量三角范畴中的扭和完全可对偶对象，证明了其可被抽象重构，并探讨了强生成元与可代表性定理的关系。
Exact weights and path metrics for triangulated categories and the derived category of persistence modules
本文在三角范畴上引入精确权重和路径度量，推广了Chuang-Lazarev秩函数，证明上同调函子诱导精确权重、三角等价诱导等距，并给出精确性的三种等价刻画，最终应用于持续模导出范畴和连续箭图表示。
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