论文标题
具有固定对象集的类别的Koszul二元性
Koszul duality for categories with a fixed object set
论文作者
论文摘要
我们定义了单型双粘合模型类别中单型物体的koszul dual的概念。我们的构造概括了经典的Yoneda代数$ EXT_A（K，K）$。我们应用这种通用结构来定义富含光谱或链复合物的类别的Koszul双重。这个示例依赖于经典观察结果，即丰富类别是一类富集图中的单体对象。我们观察到，富集图的类别是双封的，这意味着它均与左侧和右内部对象一起。给定一个类别$ r $（在古典代数中扮演地面$ k $的角色）和一个增强的$ r $ -r $ -Algebra $ c $，我们将$ c $的koszul dual of $ c $，表示为$ k（c）$，是$ r $ r $ r $ r $ - $ r $ r $ $ r $的$ r $ r $ c $ c。 我们在$ c $以上的模块类别之间建立了预期的辅助功能，而模块超过$ k（c）$。我们调查了何时从$ c $到其双重双$ k（k（c））$的地图是一个等效性的问题。我们还表明，可乐的Koszul二元性可以理解为类别Koszul双重性的特殊情况。通过这种方式，我们将Koszul二重性纳入了更广泛的背景下，并可能澄清了它的某些方面。
We define a notion of Koszul dual of a monoid object in a monoidal biclosed model category. Our construction generalizes the classic Yoneda algebra $Ext_A(k,k)$. We apply this general construction to define the Koszul dual of a category enriched over spectra or chain complexes. This example relies on the classical observation that enriched categories are monoid objects in a category of enriched graphs. We observe that the category of enriched graphs is biclosed, meaning that it comes with both left and right internal hom objects. Given a category $R$ (which plays the role of the ground field $k$ in classical algebra), and an augmented $R$-algebra $C$, we define the Koszul dual of $C$, denoted $K(C)$, as the $R$-algebra of derived endomorphisms of $R$ in the category of right $C$-modules. We establish the expected adjunctions between the categories of modules over $C$ and modules over $K(C)$. We investigate the question of when the map from $C$ to its double dual $K(K(C))$ is an equivalence. We also show that Koszul duality of operads can be understood as a special case of Koszul duality of categories. In this way we incorporate Koszul duality of operads in a wider context, and possibly clarify some aspects of it.
