Quadratic categories and Koszul resolutions

三角范畴中双正合函子与Koszul对象的生成时间

Generation time for biexact functors and Koszul objects in triangulated categories

本文研究三角范畴中度量对象生成所需锥体数的生成时间（层级），为增强三角范畴建立双正合函子与Koszul对象的不等式，并推广至增强张量三角范畴。

操作范畴的Koszul对偶

Koszul duality for operadic categories

本文为一大类操作范畴上的operads建立Koszul对偶与Koszul性理论，研究二元二次operads实例并证明其Koszul性，涵盖经典operads、cyclic、modular、wheeled operads及多种PROPs变体等结构。

Koszul duality for categories with a fixed object set

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 $…

作为Koszul对偶自然环境的操作范畴

Operadic categories as a natural environment for Koszul duality

本文为操作范畴上的operads建立Koszul对偶理论框架，提炼出支持二次operads及其Koszul性发展的附加性质，并利用离散操作（op）纤维化构造新操作范畴，重点证明了图的操作范畴满足全部附加性质。

Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence

This paper can be thought of as an extended introduction to arXiv:0708.3398; nevertheless, most of its results are not covered by loc. cit. We consider the derived categories of DG-modules, DG-comodules, and DG-contramodules, the coderived and contraderived categories of CDG-modules, the coderived categories of CDG-comodules, and the contraderived categories of CDG-contramodules. The equivalence between the latter two categories (the comodule-contramodule correspondence) is established. Nonhomogeneous Koszul duality or "triality" (an equivalence between exotic derived categories corresponding …

Koszul duality and the PBW theorem in symmetric tensor categories in positive characteristic

We generalize the theory of Koszul complexes and Koszul algebras (in particular, Koszul duality between symmetric and exterior algebras) to symmetric tensor categories. In characteristic $p\ge 5$, this theory exhibits peculiar effects, not observed in the classical theory. In particular, we show that the symmetric and exterior algebras of a non-invertible simple object in the Verlinde category ${\rm Ver}_p$ are almost Koszul (although not Koszul), and show how this gives examples of $(r,s)$-Koszul algebras with any $r,s\ge 2$. We also develop a theory of Lie algebras in symmetric tensor catego…

非分次Koszul对偶及其应用

A Non-graded Koszul Duality and Its Applications

本文建立有限维Koszul代数的有界导出范畴在分次与非分次情形下的导出Koszul对偶，无需对Koszul对偶代数施加有限性条件，并应用于范畴O的块和几何表示论中的反常层。

Koszul Feynman 范畴

Koszul Feynman Categories

本文证明了所有立方 Feynman 范畴都是 Koszul 的，并由此构造出显式极小的余纤维化解，为基于图和非基于图的广义 operad 提供 $\infty$ 版本模型。

拟阵、费曼范畴与Koszul对偶

Matroids, Feynman categories, and Koszul duality

该研究引入并研究特定费曼范畴，将拟阵的Chow环、Orlik–Solomon代数等组合不变量组装为该范畴上的操作子结构，建立二者的Koszul型对偶以推广Getzler的结果，并给出Orlik–Solomon代数组合Leray模型的新解释。

Koszul duality for compactly generated derived categories of second kind

For any dg algebra $A$ we construct a closed model category structure on dg $A$-modules such that the corresponding homotopy category is compactly generated by dg $A$-modules that are finitely generated and free over $A$ (disregarding the differential). We prove that this closed model category is Quillen equivalent to the category of comodules over a certain, possibly nonconilpotent dg coalgebra, a so-called extended bar construction of $A$. This generalises and complements certain aspects of dg Koszul duality for associative algebras.

拟阿贝尔范畴中的 Koszul 幺半群

Koszul Monoids in Quasi-abelian Categories

本文在具有足够平坦投射对象的双完备闭对称幺半拟阿贝尔范畴中，以无元素方式定义 Koszul 与二次幺半群，并借助左心嵌入证明其分次模导出范畴子范畴的对偶等价。

Enriched Koszul duality for dg categories

It is well-known that the category of small dg categories dgCat, though it is monoidal, does not form a monoidal model category. In this paper we construct a monoidal model structure on the category of pointed curved coalgebras ptdCoa* and show that the Quillen equivalence relating it to dgCat is monoidal. We also show that dgCat is a ptdCoa*-enriched model category. As a consequence, the homotopy category of dgCat is closed monoidal and is equivalent as a closed monoidal category to the homotopy category of ptdCoa*. In particular, this gives a conceptual construction of a derived internal hom…

范畴 Koszul 对偶

Categorical Koszul duality

本文建立 dg 范畴与曲余代数间的 Koszul 对偶，并用标准化链复形函子解释 dg 神经，进而刻画拟范畴表示与分层空间可构造 dg 层范畴。

通过极化得到的operad相容结构、其Koszul对偶与Manin乘积

Compatible structures of operads by polarization, their Koszul duality and Manin products

该研究将不变量理论的极化推广到operad层面，统一了线性、匹配、全相容三类operad结构，揭示了其Koszul对偶性质与Manin乘积构造，且部分有限生成二元二次operad的Koszul性在相容构造中保持。

Morita theory and singularity categories

We propose an analogue of the bounded derived category for an augmented ring spectrum, defined in terms of a notion of Noether normalization. In many cases we show this category is independent of the chosen normalization. Based on this, we define the singularity and cosingularity categories measuring the failure of regularity and coregularity and prove they are Koszul dual in the style of the BGG correspondence. Examples of interest include Koszul algebras and Ginzburg DG-algebras, $C^*(BG)$ for finite groups (or for compact Lie groups with orientable adjoint representation), cochains in ratio…

Symmetric and exterior powers of categories

We define symmetric and exterior powers of categories, fitting into categorified Koszul complexes. We discuss examples and calculate the effect of these power operations on the categorical characters of matrix 2-representations.

切触范畴中的Ehresmann联络

Ehresmann connections in tangent categories

将Ehresmann联络推广到切触范畴，给出等价刻画并证明其推广Koszul联络，定义平行移动和曲率，证明结构方程和Bianchi恒等式。