Kevin Coulembier
We determine internal characterisations for when a tensor category is (super) tannakian, for fields of positive characteristic. This generalises the corresponding characterisations in characteristic zero by P. Deligne. We also explore notions of Frobenius twists in tensor categories in positive characteristic.
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Some fast-growing tensor categories
本文利用Kuhn关于有限域一般表示理论的结果，证明有限域上半单预Tannakian范畴在特征0域上的线性化仍保持半单预Tannakian性质，并构造出增长速度显著快于已知例子的新预Tannakian范畴。
On the relationship between Galois and Tannakian categories: an open letter
本文论证域上的伽罗瓦范畴与淡中范畴并非仅形式上相似，而是淡中范畴推广了伽罗瓦范畴，且后者以结构化方式嵌入前者。
We provide conditions for a category with a fiber functor to be equivalent to the category of representations of a linear differential algebraic group. This generalizes the notion of a neutral Tannakian category used to characterize the category of representations of a linear algebraic group.
Tannakian categories: origins and summary
本文介绍Tannakian范畴理论的起源，梳理总结该领域的主要研究成果。
We classify the "quotients" of a tannakian category in which the objects of a tannakian subcategory become trivial, and we examine the properties of such quotient categories.
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.
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…
Pre-Galois categories and Fraïssé's theorem
本文引入预伽罗瓦范畴作为预坦纳卡范畴的组合类比，证明寡态群的光滑G-集范畴构成预伽罗瓦范畴且此类例子穷尽，从而重新表述弗拉伊塞定理，并借助置换类给出非预伽罗瓦的B-范畴例子。
We introduce and develop the notion of scalar extension for abelian categories. Given a field extension F'/F, to every F-linear abelian category A satisfying a suitable finiteness condition we associate an F'-linear abelian category A' and an exact F-linear functor t: A --> A'. This functor is universal among F-linear right exact functors with target an F'-linear abelian category. We discuss various basic properties of this concept, among others compatibilities with multilinear endofunctors such as tensor products, and the permanence of favourable properties of the functors and categories invo…
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…
Strong similarities have been long observed between the Galois (Categories Galoisiennes) and the Tannaka (Categories Tannakiennes) theories of representation of groups. In this paper we construct an explicit (neutral) Tannakian context for the Galois theory of atomic topoi, and prove the equivalence between its fundamental theorems. Since the theorem is known for the Galois context, this yields, in particular, a proof of the fundamental (recognition) theorem for a new Tannakian context. This example is different from the additive cases or their generalization, where the theorem is known to hol…
We study several classes of braided fusion categories, and prove that they all contain nontrivial Tannakian subcategories. As applications, we classify some fusion categories in terms of solvability and group-theoreticality.
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…
We prove an analog of the Künneth formula for the groups of minimal non-degenerate extensions arXiv:1602.05936 of symmetric fusion categories. We describe in detail the structure of the group of minimal extensions of a pointed super-Tannakian fusion category. This description resembles that of the third cohomology group of a finite abelian group. We explicitly compute this group in several concrete examples.
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.
For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties - expressible entirely within the language of tensor categories - which are necessary and suf…
We introduce, for a symmetric fusion category $\mathcal{A}$ with Drinfeld centre $\mathcal{Z}(\mathcal{A})$, the notion of $\mathcal{Z}(\mathcal{A})$-crossed braided tensor category. These are categories that are enriched over $\mathcal{Z}(\mathcal{A})$ equipped with a symmetric tensor product, while being braided monoidal with respect to the usual tensor product on $\mathcal{Z}(\mathcal{A})$. In the Tannakian case where $\mathcal{A}=\mathbf{Rep}(G)$ for a finite group $G$, the 2-category of $\mathcal{Z}(\mathcal{A})$-crossed braided categories is shown to be equivalent to the 2-category of $G…
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.
Under certain conditions, a scheme can be reconstructed from its category of quasi-coherent sheaves. The Tannakian reconstruction theorem provides another example where a geometric object can be reconstructed from an associated category, in this case the category of its finite dimensional representations. Lurie's result that the pseudofunctor which sends a geometric stack to its category of quasi-coherent sheaves is fully faithful provides a conceptual explanation for why this works. In this paper we prove a generalized Tannakian recognition theorem, in order to characterize a part of the imag…
We study the question when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.
We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,d) depending on a degree function d. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows: 1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories. 2. Using lattice theory, we give a simple numerical criterion for the vanishing of N. 3. We determine all degree functions for w…
A natural question in the theory of Tannakian categories is: What if you don't remember $\Forget$? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid $π_1(\spec(R))$, i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for étale $π_1(\spec(R))$ in terms of the category of $R$-modules rather than the category of étale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric mo…
Let k be any field. J-P. Serre proved that the spectrum of the Grothendieck ring of the k-representation category of a group is connected, and that the same holds in characteristic zero for the representation category of a Lie algebra over k. We say that a tensor category C over k is virtually indecomposable if its Grothendieck ring contains no nontrivial central idempotents. We prove that the following tensor categories are virtually indecomposable: Tensor categories with the Chevalley property; representation categories of affine group schemes; representation categories of formal groups; rep…
We investigate the perfect derived category dgPer(A) of a positively graded differential graded (dg) algebra A whose degree zero part is a dg subalgebra and semisimple as a ring. We introduce an equivalent subcategory of dgPer(A) whose objects are easy to describe, define a t-structure on dgPer(A) and study its heart. We show that dgPer(A) is a Krull-Remak-Schmidt category. Then we consider the heart in the case that A is a Koszul ring with differential zero satisfying some finiteness conditions.
For any positive integer $n$, $n$-derived-simple derived discrete algebras are classified up to derived equivalence. Furthermore, the Jordan-Hölder theorems for all kinds of derived categories of derived discrete algebras are obtained.
A Tannakian category is an abelian tensor category equipped with a fiber functor and additional structures which ensure that it is equivalent to the category of representations of some affine groupoid scheme acting on the spectrum of a field extension. If we are working over an arbitrary commutative ring rather than a field, the categories of representations cease to be abelian. We provide a list of sufficient conditions which ensure that an additive tensor category is equivalent to the category of representations of an affine groupoid scheme acting on an affine scheme, or, more generally, to …
Singularity Categories of Simple Singularities in Positive Characteristic
本文研究正特征下同维简单奇点的奇异范畴，推广Hua与Keller的相关定理，确定有理二重点奇异范畴为标准的条件，完善了正特征下奇点范畴的相关理论。
We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda's theory of higher extensions to this framework. We show that, when there are enough projectives or enough injectives, thus defined extensions agree with the ones defined earlier via projective or injective resolutions. For categories with enough projective or enough injective morphisms, we prove that these are right derived functors of the $\operatorname{Hom}$-bifunctor in either argument. Since $\operatorname{Hom}$ is only half-exact in each argument, it is natural to expec…
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 Leinster matrix corresponding to a finite category has entries counting the number of morphisms between objects. A first question is to know which positive integer matrices come from at least one finite category. Here, that question will be answered when the entries are strictly positive.
We show that direct summands of certain additive functors arising as bifunctors with a fixed argument in an abelian category are again of that form whenever the fixed argument has finite length or, more generally, satisfies the descending chain condition on images of nested endomorphisms. In particular, this provides a positive answer to a conjecture of M. Auslander in the case of categories of finite modules over artin algebras. This implies that the covariant Ext functors are the only injectives in the category of defect-zero finitely presented functors on such categories.
Monoidal abelian envelopes
本文证明了非阿贝尔幺半群范畴的阿贝尔包络的构造性存在定理，为张量范畴的构造提供新工具，并给出Deligne猜想及正特征下张量范畴的新证明。
We propose a conjectural extension to positive characteristic case of a well known Deligne's theorem on the existence of super fiber functors. We prove our conjecture in the special case of semisimple categories with finitely many isomorphism classes of simple objects.
N-spherical functors and tensor categories
本文将稳定无穷范畴的 $N$-球面函子引入幺半范畴，定义 $N$-有界对象，并揭示其与 Jones-Wenzl 幂等元、Frobenius-Perron 维数及正特征对称张量范畴核心猜想的联系。
We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $\bf k$. If ${\rm char}({\bf k})=p>0$, we use this method to construct generalizations ${\rm Ver}_{p^n}$, ${\rm Ver}_{p^n}^+$ of the incompressible abelian symmetric tensor categories defined in arXiv:1807.05549 for $p=2$ and by Gelfand-Kazhdan and Georgiev-Mathieu for $n=1$. Namely, ${\rm Ver}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for $SL_2(\bf k)$ by the $n$-th Steinberg module, and ${\rm Ver}_{p^n}^+$ is its sub…
We prove that the bounded derived category of the incidence algebra of the Tamari lattice is fractionally Calabi-Yau, giving a positive answer to a conjecture of Chapoton. The proof involves a combinatorial description of the Serre functor of this derived category on a sufficiently nice family of indecomposable objects.
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…
Singularity categories of rational double points in arbitrary characteristic
本文在任意特征下建立了有理二重点的奇点范畴与单边laced Dynkin图的一一对应，并利用正特征下非taut的有理二重点构造了Hua-Keller定理的反例。
We introduce the framework of AECats (abstract elementary categories), generalising both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory ("cat", as introduced by Ben-Yaacov) forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of (subsets of) models of a positive or continuous theory is an AECat. The Kim-Pillay theorem for first-order logic characterises simple theories by the properties dividing independence has. We prove a version of the Kim-Pillay theorem for …
Closure and Interior Operators of the Category of Positive Topologies
本文定义了正拓扑中收敛覆盖范畴上的闭包与内部算子，构造了相应的具体范畴并证明其为拓扑范畴。
We introduce, for every positive integer n, the notion of an n-relative category and show that the category of the small n-relative categories is a model for the homotopy theory of n-fold homotopy theories, i.e. homotopy theories of ... of homotopy theories.
We classify semisimple module categories over the tensor category of representations of quantum SL(2) extending previous results to the roots of unity and positive characteristic cases.
Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $G$ be an affine group scheme over $k$. We classify the indecomposable exact module categories over the rigid tensor category $\text{Coh}_f(G)$ of coherent sheaves of finite dimensional $k-$vector spaces on $G$, in terms of $(H,ψ)-$equivariant coherent sheaves on $G$. We deduce from it the classification of indecomposable {\em geometrical} module categories over $\Rep(G)$. When $G$ is finite, this yields the classification of {\em all} indecomposable exact module categories over the finite tensor category $\Rep(G)$. In pa…
We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and classify module categories over the fusion category of $\hat{sl}(2)$ at a positive integer level where we meet once again ADE classification pattern.
We develop the notion of deformation of a morphism in a left-proper model category. As an application we provide a geometric/homotopic description of deformations of commutative (non-positively) graded differential algebras over a local DG-Artin ring.
The strict globular $ω$-categories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) $ω$-category $\C$ three homology theories. The first one is called the globular homology. It contains the oriented loops of $\C$. The two other ones are called the negative (resp. positive) corner homology. They contain in a certain manner the branching areas of execution paths or negative corners (resp. the merging areas of execution paths or positive corners) of $\C$. Two natural linear maps called the negative (resp. the positive) Hurewic…
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…
We give necessary and sufficient conditions for stratification and costratification to descend along a coproduct preserving, tensor-exact $R$-linear functor between $R$-linear tensor-triangulated categories which are rigidly-compactly generated by their tensor units. We then apply these results to non-positive commutative DG-rings and connective ring spectra. In particular, this gives a support-theoretic classification of (co)localizing subcategories, and thick subcategories of compact objects of the derived category of a non-positive commutative DG-ring with finite amplitude, and provides a f…
We introduce the CP*-construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive maps, the CP*-construction forms its category of "abstract C*-algebras" and completely positive maps. This analogy is justified by the case of finite-dimensional Hilbert spaces, where the CP*-construction yields the category of finite-dimensional C*-algebras and completely positive maps. The CP*-construction fully embeds Selinger's CPM-constru…
A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant $\mathcal{C}'$ of a fully faithful representation $\mathcal{C}\to\operatorname{Bim}(R)$ of a unitary fusion category $\mathcal{C}$. Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent unitary fusion categories, then their commutant categories $\mathcal{C}'$ and $\mathcal{D}'$ are equivalent as bicommutant …
Any refinement system (= functor) has a fully faithful representation in the refinement system of presheaves, by interpreting types as relative slice categories, and refinement types as presheaves over those categories. Motivated by an analogy between side effects in programming and *context effects* in linear logic, we study logical aspects of this "positive" (covariant) representation, as well as of an associated "negative" (contravariant) representation. We establish several preservation properties for these representations, including a generalization of Day's embedding theorem for monoidal…
We give new characterisations of regular Mal'tsev categories with distributive lattice of equivalence relations through variations of the so-called Triangular Lemma and Trapezoid Lemma in universal algebra. We then give new characterisations of equivalence distributive Goursat categories (which extend $3$-permutable varieties) through variations of the Triangular and Trapezoid Lemmas involving reflexive and positive relations.
We formulate and study Howe-Moore type properties in the setting of quantum groups and in the setting of rigid $C^{\ast}$-tensor categories. We say that a rigid $C^{\ast}$-tensor category $\mathcal{C}$ has the Howe-Moore property if every completely positive multiplier on $\mathcal{C}$ has a limit at infinity. We prove that the representation categories of $q$-deformations of connected compact simple Lie groups with trivial center satisfy the Howe-Moore property. As an immediate consequence, we deduce the Howe-Moore property for Temperley-Lieb-Jones standard invariants with principal graph $A_…
Categories of relations over a regular category form a family of models of quantum theory. Using regular logic, many properties of relations over sets lift to these models, including the correspondence between Frobenius structures and internal groupoids. Over compact Hausdorff spaces, this lifting gives continuous symmetric encryption. Over a regular Mal'cev category, this correspondence gives a characterization of categories of completely positive maps, enabling the formulation of quantum features. These models are closer to Hilbert spaces than relations over sets in several respects: Heisenb…
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