Mostafa Einollahzadeh, Amir Jafari
In this paper we generalize Tannakian formalism to fiber functors over general tensor categories. We will show that (under some technical conditions) if the fiber functor has a section, then the source category is equivalent to the category of comodules over a Hopf algebra in the target category. We will also give a description of this Hopf algebra using the notion of framed objects.
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We consider essentially small rigid tensor categories (not necessarily abelian) which have a faithful tensor functor to a category of super vector spaces over a field of characteristic 0. It is shown how to construct for each such tensor category a super Tannakian hull, which is a universal faithful tensor functor to a super Tannakian category over a field of characteristic 0. The construction is analogous to the passage from an integral domain to its field of fractions.
The principal aim of this note is to give an elementary proof of the fact that any two fiber functors of a Tannakian category are locally isomorphic. This builds on an idea of Deligne concerning scalar extensions of Tannakian categories and implements a proof strategy which Deligne attributes to Grothendieck. Besides categorical generalities, the proof merely relies on basic properties of exterior powers and the classification of finitely generated modules over a principal ideal domain. Using related ideas (but less elementary means) we also present an alternative characterization of Tannakian…
Scalar Extension of Abelian and Tannakian Categories
我们引入并发展了阿贝尔范畴的标量扩张的概念。给定一个域扩张F '/F，对于每个满足适当有限性条件的F-线性阿贝尔范畴A，我们将一个F'-线性阿贝尔范畴A'和一个正合F-线性函子t：A --> A'联系起来.这个函子在以F '-线性阿贝尔范畴为目标的F-线性右正合函子中是泛函子。我们讨论了这个概念的各种基本性质，其中包括与多线性内函子（例如张量积）的兼容性，以及所涉及的函子和范畴的有利性质的持久性。我们得到了Tannakian范畴的标量扩张的概念，这使我们能够推导出Tannakian范畴的代数单值群的结果。
Given a rigid tensor-triangulated category and a vector space valued homological functor for which the Künneth isomorphism holds, we construct a universal graded-Tannakian category through which the given homological functor factors. We use this to (unconditionally) construct graded-Tannakian categories of pure motives associated to a fixed Weil cohomology theory, with a fiber functor realizing the given cohomology theory. For $\ell$-adic cohomology and a ground field which is algebraic over a finite field, this category is Tannakian. In this case, we obtain in particular motivic Galois groups…
We establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring. We use this duality and the known Tannakian duality due to Saavedra to study morphisms between flat affine group schemes. Next, we apply our new duality to the category of stratified sheaves on a smooth scheme over a Dedekind ring R to define the relative differential fundamental group scheme of the given scheme and compare the fibers of this group scheme with the fundamental group scheme of the fibers. When R …
Tannakian categories, linear differential algebraic groups, and parameterized linear differential equations
本文给出了一个具有纤维函子的范畴等价于一个线性微分代数群的表示范畴的条件。这推广了中立Tannakian范畴的概念，用来描述线性代数群的表示范畴。
We explain why every non-trivial exact tensor functor on the triangulated category of mixed motives over a field F has zero kernel, if one assumes "all" motivic conjectures. In other words, every non-zero motive generates the whole category up to the tensor triangulated structure. Under the same assumptions, we also give a complete classification of triangulated étale motives over F with integral coefficients, up to the tensor triangulated structure, in terms of the characteristic and the orderings of F.
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic p version of this theorem. Namely we show that a pre-Tannakian category over an algebraically closed field of characteristic p>0 admits a fiber functor into the Verlinde…
Which abelian tensor categories are geometric?
对于一个大类的几何对象，准凝聚层范畴的通道提供了一个嵌入在阿贝尔张量范畴的2-范畴。作者引入的弱Tannakian范畴的概念给出了这种嵌入图像中张量范畴的特征。然而，这个概念需要额外的结构存在，即纤维函子。对于特征为零的经典Tannakian范畴的情况，Deligne发现了内在性质-完全可以在张量范畴的语言中表达-这是纤维函子存在的必要和充分条件。本文将Deligne的结果推广到特征为零的弱Tannakian范畴。可以用这种方式识别准相干层的张量范畴的几何对象类包括经典Tannaka对偶中产生的格贝和更经典的几何对象，如特征为零的域上的投影簇。我们的证明使用不同的角度纤维函子，我们通过几何张量范畴的概念形式化。这个观点的第二个应用使我们能够描述纤维积上的准相干层的种类。
The fundamental pro-groupoid of an affine 2-scheme
坦纳基范畴理论中的一个自然问题是：如果你不记得$\Forget$了怎么办？在任意交换环$R$上，我们证明了这个问题的一个答案是由等价基本广群$π_1(\spec(R))$表示的函子给出的，即$R$的可分绝对伽罗瓦群，当它是一个域时。这就用$R$ -模范畴而不是用étale覆盖范畴给出了étale $π_1(\spec(R))$的新定义。更一般地，我们引入了一个新的概念“交换2-环”，它包括Grothendieck拓扑和对称幺半范畴的模，并定义了一个概念，$π_1$相应的“仿射2-计划。“这些结果有助于简化和澄清étale基本群的一些特性。例如，étale基本群不是“真”群，而仅仅投射有限群，并且人们不能希望恢复更多：由一个概型的étale基本群表示的“塔纳基”函子保持有限乘积，但不是所有乘积。
Given a symmetric monoidal stable $\infty$-category $\mathcal{C}$ and a left adjoint symmetric monoidal fiber functor to $\operatorname{Mod}_A^{\otimes}$ for some $\mathbb{E}_{\infty}$-ring $A$, one can construct a derived group scheme $G$ of monoidal automorphisms of this functor. The left adjoint fiber functor also induces a monad on $\mathcal{C}$. Under some finiteness hypothesis on the fiber functor, we show there is a comparison functor from the category of representations of $G$ to the descent category of the induced monad on $\mathcal{C}$.
Fiber Functors on Temperley-Lieb Categories
利用非退化双线性型的分类结果研究了Temperley-Lieb范畴上的Fiber函子.酉纤维函子的情况下也被认为是。
A theory of a derivator version of six-functor-formalisms is developed, using an extension of the notion of fibered multiderivator due to the author. Using the language of (op)fibrations of 2-multicategories this has (like a usual fibered multiderivator) a very neat definition. This definition not only encodes all compatibilities among the six functors but also their interplay with homotopy Kan extensions. One could say: a nine-functor-formalism. This is essential for dealing with (co)descent questions. Finally, it is shown that every fibered multiderivator (for example encoding any kind of de…
Bundle functors and fibrations
我们给出了一个帐户，在纤维范畴和他们的纤维状，方面的理论丛函子和星丛函子在微分几何。
Tannaka Theory for Topos
我们认为局部域$B$是超格张量范畴$s\ell$中的代数。我们证明了伽罗瓦理论中开局部满射$sh(B) \stackrel{q}{\longrightarrow} \mathcal{E}$的Joyal-Tierney下降定理[Grothendieck的伽罗瓦理论的扩展，AMS Memoirs 151]与$s\ell$上的Tannakian识别定理之间的等价性，其中$s\ell$ -函子$Rel(E) \stackrel{Rel(q^*)}{\longrightarrow} Rel(sh(B)) \cong (B$ - $Mod)_0$进入离散$B$ -模的$s\ell$ -范畴。从而得到了一个新的Tannaka识别定理，它与迄今已知的识别定理有本质的不同。这种等价性来自两个独立的结果。本文给出了Joyal-Tierney与$q$相关联的局部广群$G$的显式构造，并与与$Rel(q^*)$相关联的Hopf代数体$L$的DeligneTannakian构造作了详尽的比较，证明了它们是同构的，即$L \cong \mathcal{O}(G)$ .另一方面，我们证明了任何局部广群$G$的分类拓扑的关系的$s\ell$ -范畴等价于$L$ -余模与离散子模$B$ -模的$s\ell$ -范畴，其中$L = \mathcal{O}(G)$ .我们被迫工作在一个任意的基础拓扑，因为相反的中性情况…
We lay the foundations for a model theoretic study of proalgebraic groups. Our axiomatization is based on the tannakian philosophy. Through a tensor analog of skeletal categories we are able to consider neutral tannakian categories with a fibre functor as many-sorted first order structures. The class of diagonalizable proalgebraic groups is analyzed in detail. We show that the theory of a diagonalizable proalgebraic group $G$ is determined by the theory of the base field and the theory of the character group of $G$. Some initial steps towards a comprehensive study of types are also made.
Fiber functors, monoidal sites and Tannaka duality for bialgebroids
小可加幺半范畴C上的非交换纤维函子是什么？我们给出了一个答案，导致一个新的Tannaka对偶定理bialgebroids推广以前的结果Phung何海。这个构造揭示了一个层理论的解释，只要重构的双代数胚H具有等价于T-层范畴的余模范畴。C上的么半群Grothendieck拓扑.我们还证明了具有有界融合和弱核的小可加幺半范畴范畴上纤维函子的存在性定理。对于某些自治范畴，一个广义的Ulbrich定理可以被公式化，它将纤维函子与Hopf代数体伽罗瓦扩张联系起来。
We develop the theory of (op)fibrations of 2-multicategories and use it to define abstract six-functor-formalisms. We also give axioms for Wirthmüller and Grothendieck formalisms (where either $f^!=f^*$ or $f_!=f_*$) or intermediate formalisms where we have e.g. a natural morphism $f_! \rightarrow f_*$. Finally, it is shown that a fibered multiderivator (in particular, a closed monoidal derivator) can be interpreted as a six-functor-formalism on diagrams (small categories). This gives, among other things, a considerable simplification of the axioms and of the proofs of basic properties, and cl…
How to build a Hopf algebra
构造了一个从满足附加性条件的 $(\infty,3)$-范畴收缩到辫子幺半 $(\infty,1)$-范畴中 Hopf 代数的函子，为 Tannaka 重建提供了高阶范畴论解释。
Conditional flatness, fiberwise localizations, and admissible reflections
本文将条件平坦性从群论推广到带点范畴，证明理想确定范畴的Birkhoff子范畴产生条件平坦局部化，并给出判定无挠反射的准则，适用于半阿贝尔簇中的零化函子。
Categorical Symmetries and Fiber Functors from Multiple Gaugeable Homomorphisms from 6D ${\cal N}=(2,0)$ SCFTs
利用SymTFT/TO对应和6D SCFT的高阶范畴结构，证明内在非可逆对称性的相对可规范代数总量子维度更大，支持超选择扇区重数可超1。
Fibrations as Eilenberg-Moore algebras
本文用单子和2-范畴的概念对纤维范畴（或者更确切地说是纤维范畴）的一些基本事实作了初步的阐述。该帐户避免任何提及范畴值函子和伪函子。
On the functoriality of universal abelian factorizations
本文研究箭图表示的通用 Abel 分解在正合函子、自然变换及一般范畴构造下的函子性，并将其推广到多重线性函子与 Abel 纤维范畴，进而处理幺半结构并应用于 perverse Nori motives。
Ind-abelian categories and quasi-coherent sheaves
研究了当一个ind-对象范畴是阿贝尔范畴时的问题。我们的答案允许进一步推广的概念弱Tannakian范畴介绍了作者。作为应用，我们证明了在适当的条件下，具有分解性质的两个格式的乘积上的相干层范畴由两个因子的相干层范畴的Deligne张量积给出.为此，我们证明了具有分解性质的拟紧半分离格式类在纤维积下是闭的。
From fibered symmetric bimonoidal categories to symmetric spectra
本文定义了纤维对称双单范畴的概念。粗略地说，这些是纤维范畴D-> C，其纤维是由C参数化的对称幺半范畴，并且使得D和C都具有满足我们所描述的某些凝聚性的对称幺半范畴的进一步结构。我们的目标是证明我们可以用函子的方式对应于纤维对称双单类范畴的E_{\infty}-环谱。
We develop a theory of Frobenius functors for symmetric tensor categories (STC) $\mathcal{C}$ over a field $\bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor $F: \mathcal{C}\to \mathcal{C}\boxtimes {\rm Ver}_p$, where ${\rm Ver}_p$ is the Verlinde category (the semisimplification of ${\rm Rep}_{\bf k}(\mathbb{Z}/p)$). This generalizes the usual Frobenius twist functor in modular representation theory and also one defined in arXiv:1503.01492, where it is used to show that if $\mathcal{C}$ i…
Tensor categories (after P. Deligne)
这些注记给出了Deligne关于超纤维函子的超张力定理的一个说明。
Quasisymmetric and unipotent tensor categories
我们对C上具有指数增长性的拟对称辫状张量范畴进行分类，即，正方形编织是任何两个简单物体的乘积的恒等式。这推广了Deligne关于指数增长性对称范畴的分类结果和Drinfeld关于拟三角拟霍普夫代数的分类结果。特别地，我们对幂单的指数增长性的辫状范畴进行分类，即，唯一的简单对象是单位对象的那些。我们还对纤维函子进行了分类。最后，利用Poisson代数群的Etingof-Kazhdan量子化理论，给出了上连通霍普夫代数的一个分类，即具有纤维函子的指数增长性的幂单范畴的分类.
Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories
本文定义并刻画群论融合 2-范畴，分析其结构、融合规则与分次，并分类带 fiber 2-函子及 Tambara-Yamagami 缺陷的融合 2-范畴。
The formal theory of Tannaka duality
Tannakian范畴是一个阿贝尔张量范畴，它配备了一个纤维函子和额外的结构，这些结构确保它等价于作用在域扩张的谱上的某个仿射广群概型的表示范畴。如果我们研究的是任意的交换环而不是域，则表示范畴不再是阿贝尔的。我们提供了一个充分条件的列表，以确保添加剂张量范畴是等价的仿射广群计划的表示的范畴上的仿射计划，或更一般地说，在一个对称的幺半范畴范畴的Hopf代数体的表示范畴。为了做到这一点，我们开发了一个“Tannaka二元性的形式理论”的启发罗斯街的“单子的形式理论。”“我们将我们的结果应用于某些类别的过滤模块，用于研究p-adic伽罗瓦表示。
Given a Noetherian formal scheme $\hat X$ over ${\rm Spf}(R)$, where $R$ is a complete DVR, we first prove a theorem of meromorphic descent along a possibly infinite cover of $\hat{X}$. Using this we construct a specialization functor from the category of continuous representations of the pro-étale fundamental group of the special fiber to the category of $F$-divided sheaves on the generic fiber. This specialization functor partially recovers the specialization functor of the étale fundamental groups. We also express the pro-étale fundamental group of a connected scheme $X$ of finite type over…
Matrix factorizations for nonaffine LG-models
我们提出了一类非仿射Landau-Ginzburg模型的矩阵分解的自然定义。对于任何LG-模型，我们构造了一个完全忠实函子，从以这种方式定义的矩阵分解的范畴到相应纤维的奇点的三角化范畴。我们还证明了，如果LG-模型的全空间是光滑的，这个函子是一个等价的。
The tensor triangular geometry of fully faithful functors
该研究证明全忠实几何函子诱导的Balmer谱映射是纤维连通的商映射，引入tt-范畴的集中与幺元化概念，计算相关范畴的Balmer谱并研究比较映射性质，给出理解tt-范畴谱的分层方法。
Reducing and Classifying Fiber Bundles over Small Categories
将小范畴上的纤维丛视为到小范畴范畴的局部常值函子，通过Grothendieck构造得到具有双纤维化投影的全范畴，并利用单值性分类纤维丛，证明规范群同构于单值子群的中心化子。
Biset functors for categories
本文将群上的双集函子推广到任意有限范畴，建立 Burnside 环、简单函子、幺半结构与纤维化等理论，并通过 Boolean 格联系对应函子。
We use the terms $\infty$-categories and $\infty$-functors to mean the objects and morphisms in an $\infty$-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of fibrant objects. Quasi-categories, Segal categories, complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, $θ_n$-spaces, and fibered versions of each of these are all $\infty$-categories in this sense. Previous work in this series shows that the basic category theory of $\infty$-categories and $\infty$-functors can be developed only in reference to the axioms …
Homotopy limits of model categories and more general homotopy theories
推广了模型范畴的同伦纤维积的定义，给出了模型范畴间左Quillen函子图的同伦极限的定义。正如前面已经表明的同伦纤维积，我们证明，这样的同伦极限实际上对应于通常的同伦极限，当我们在一个更一般的模型同伦理论，其中它们可以被视为一个模型范畴的对象。
Fiberwise building and stratification in tensor triangular geometry
本文研究了在具有小余积的张量三角范畴之间的一族保持余积的tt-函子的条件，确保由对象x∈T生成的局部化张量理想由那些在每个f_i下图像位于由f_i(x)生成的局部化张量理想中的对象决定。这导致了在刚性紧生成的张量三角范畴中层状结构的纤维化准则。作为应用，作者证明了有限群在任意Noetherian基上的排列模的大导出范畴是层状的。此外，本文的方法扩展到有限群方案在Noetherian基上的表示范畴，从而恢复了文献中最近的结果。
Reflexivity and dualizability in categorified linear algebra
上完全线性范畴$\mathcal C$的“线性对偶”是所有上连续线性函子$\mathcal C \to \mathrm{Vect}$的范畴。我们研究了当一个上完全线性范畴是自反的（等价于它的双重对偶）或可对偶的（与它的对偶的配对伴随着相应的配对）。我们的主要结果是，范畴的余模的可数维余结合余代数总是自反的，但（没有任何维数假设）dualizable当且仅当它有足够的射影，这很少发生。在此过程中，我们证明了栈$X$上的拟相干层范畴$\mathrm{Qcoh}(X)$是不可对偶的，如果$X$是正特征的半单代数群的分类栈，或者如果$X$是包含正维数的闭投射子概型的概型，但是如果$X$是仿射概型与虚线性约化群的商，则是可对偶的。最后，我们证明了具有可数指标偏序集的仿射ind-方案的张量性（一种Tannakian对偶）。
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