Fiber Functors Research Articles

On Frobenius exact symmetric tensor categories

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 category $\mathrm{Ver}_p$ (i.e., is the representation category of an affine group scheme in $\mathrm{Ver}_p$) if and only if it has moderate growth and is Frobenius exact. This implies that Frobenius exact pre-Tannakian categories of moderate growth admit a well behaved notion of Frobenius-Perron dimension. It follows that any semisimple pre-Tannakian category of moderategrowth has a fiber functor to $\mathrm{Ver}_p$ (so in particular Deligne's theorem holds on the nose for semisimple pre-Tannakian categories in characteristics $2$,$3$). This settles a conjecture of the third author from 2015. In particular, this result applies to semisimplifications of categories of modular representations of finite groups (or, more generally, affine group schemes), which gives new applications to classical modular representation theory. For example, it allows us to characterize, for a modular representation $V$, the possible growth rates of the number of indecomposable summands in $V^{\otimes n}$of dimension prime to $p$.

New incompressible symmetric tensor categories in positive characteristic

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field k. If char(k)=p>0, then we use this method to construct the incompressible abelian symmetric tensor categories Verpn, Verpn+ generalizing earlier constructions by Gelfand–Kazhdan and Georgiev–Mathieu for n=1, and by Benson–Etingof for p=2. Namely, Verpn is the abelian envelope of the quotient of the category of tilting modules for SL2(k) by the nth Steinberg module, and Verpn+ is its subcategory generated by PGL2(k)-modules. We show that Verpn are reductions to characteristic p of Verlinde braided tensor categories in characteristic 0, which explains the notation. We study the structure of these categories in detail and, in particular, show that they categorify the real cyclotomic rings Z[2cos(2π∕pn)], and that Verpn embeds into Verpn+1. We conjecture that every symmetric tensor category of moderate growth over k admits a fiber functor to the union Verp∞ of the nested sequence Verp⊂Verp2⊂⋯ . This would provide an analogue of Deligne’s theorem in characteristic 0 and a generalization of the results of Coulembier, Etingof, and Ostrik, which shows that this conjecture holds for Frobenius exact (in particular, semisimple) categories, and, moreover, the fiber functor lands in Verp (in the case of fusion categories, this was shown earlier by Ostrik). Finally, we classify symmetric tensor categories generated by an object with invertible exterior square; this class contains the categories Verpn.

Étale Covers and Fundamental Groups of Schematic Finite Spaces

We introduce the category of finite étale covers of an arbitrary schematic space X and show that, equipped with an appropriate natural fiber functor, it is a Galois Category. This allows us to define the étale fundamental group of schematic spaces. If X is a finite model of a scheme S, we show that the resulting Galois theory on X coincides with the classical theory of finite étale covers on S, and therefore, we recover the classical étale fundamental group introduced by Grothendieck. To prove these results, it is crucial to find a suitable geometric notion of connectedness for schematic spaces and also to study their geometric points. We achieve these goals by means of the strong cohomological constraints enjoyed by schematic spaces.

Unitary transformations of fibre functors

We study unitary pseudonatural transformations (UPTs) between fibre functors Rep(G) -> Hilb, where G is a compact quantum group. For fibre functors F_1, F_2 we show that the category of UPTs F_1 -> F_2 and modifications is isomorphic to the category of finite-dimensional *-representations of the corresponding bi-Hopf-Galois object. We give a constructive classification of fibre functors accessible by a UPT from the canonical fibre functor, as well as UPTs themselves, in terms of Frobenius algebras in the category Rep(A_G), where A_G is the Hopf *-algebra dual to the compact quantum group. As an example, we show that finite-dimensional quantum isomorphisms from a quantum graph X are UPTs between fibre functors on Rep(G_X), where G_X is the quantum automorphism group of X.

A group-theoretic generalization of thep-adic local monodromy theorem

AbstractLetGbe a connected reductive group over ap-adic number fieldF. We propose and study the notions ofG-$\varphi $-modules andG-$(\varphi ,\nabla )$-modules over the Robba ring, which are exact faithfulF-linear tensor functors from the category ofG-representations on finite-dimensionalF-vector spaces to the categories of$\varphi $-modules and$(\varphi ,\nabla )$-modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlaya’s slope filtration theorem in this context, and show thatG-$(\varphi ,\nabla )$-modules over the Robba ring are “G-quasi-unipotent,” which is a generalization of thep-adic local monodromy theorem proved independently by Y. André, K. S. Kedlaya, and Z. Mebkhout.

Topological field theories and symmetry protected topological phases with fusion category symmetries

Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples (Z, M, i, s, ϕ) where (Z, M, i) is a fiber functor, s is a sign, and ϕ is the action of orientation- reversing symmetry that is compatible with the fiber functor (Z, M, i). We apply this classification to SPT phases with Kramers-Wannier-like self-duality.

Triangle presentations and tilting modules for SL$_{2k+1}$

Triangle presentations are combinatorial structures on finite projective geometries which characterize groups acting simply transitively on the vertices of a locally finite building of type \tilde{A}_{n-1},\ (n \ge 3) . From a type \tilde{A}_{n-1} triangle presentation on a geometry of order q , we construct a fiber functor on the diagrammatic monoidal category \mathrm{Web}(\mathrm{SL}^{-}_{n}) over any field \mathbb{k} with characteristic p\ge n − 1 such that q \equiv 1\ \mathrm{mod}\ p . When \mathbb{k} is algebraically closed and n odd, this gives new fiber functors on the category of tilting modules for \mathrm{SL}_{n} .

Model theory of proalgebraic groups

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 G is determined by the theory of the base field and the theory of the character group of G G . Some initial steps towards a comprehensive study of types are also made.

The intertwiner spaces of non-easy group-theoretical quantum groups

In 2015, Raum and Weber gave a definition of group-theoretical quantum groups, a class of compact matrix quantum groups with a certain presentation as semi-direct product quantum groups, and studied the case of easy quantum groups. In this article we determine the intertwiner spaces of non-easy group-theoretical quantum groups. We generalise group-theoretical categories of partitions and use a fiber functor to map partitions to linear maps which is slightly different from the one for easy quantum groups.We show that this construction provides the intertwiner spaces of group-theoretical quantum groups in general.

On the Frobenius functor for symmetric tensor categories in positive characteristic

Abstract We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 {\mathcal{C}} over a field 𝒌 {\boldsymbol{k}} of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : 𝒞 → 𝒞 ⊠ Ver p {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}} , where Ver p {{\rm Ver}_{p}} is the Verlinde category (the semisimplification of Rep 𝐤 ( ℤ / p ) {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{Z}/p)} ); a similar construction of the underlying additive functor appeared independently in [K. Coulembier, Tannakian categories in positive characteristic, preprint 2019]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [V. Ostrik, On symmetric fusion categories in positive characteristic, Selecta Math. (N.S.) 26 2020, 3, Paper No. 36], where it is used to show that if 𝒞 {\mathcal{C}} is finite and semisimple, then it admits a fiber functor to Ver p {{\rm Ver}_{p}} . The main new feature is that when 𝒞 {\mathcal{C}} is not semisimple, F need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor 𝒞 → Ver p {\mathcal{C}\to{\rm Ver}_{p}} . We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of F, and use it to show that for categories with finitely many simple objects F does not increase the Frobenius–Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which F is exact, and define the canonical maximal Frobenius exact subcategory 𝒞 ex {\mathcal{C}_{\rm ex}} inside any STC 𝒞 {\mathcal{C}} with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius–Perron dimension is preserved by F. One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to Ver p {{\rm Ver}_{p}} . This is the strongest currently available characteristic p version of Deligne’s theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of F lands in 𝒞 ex {\mathcal{C}_{\rm ex}} . Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra 𝒌 ⁢ [ d ] / d 2 {\boldsymbol{k}[d]/d^{2}} with d primitive and R-matrix R = 1 ⊗ 1 + d ⊗ d {R=1\otimes 1+d\otimes d} ), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [P. Etingof and S. Gelaki, Exact sequences of tensor categories with respect to a module category, Adv. Math. 308 2017, 1187–1208].

Finite symmetric tensor categories with the Chevalley property in characteristic 2

We prove an analog of Deligne’s theorem for finite symmetric tensor categories [Formula: see text] with the Chevalley property over an algebraically closed field [Formula: see text] of characteristic [Formula: see text]. Namely, we prove that every such category [Formula: see text] admits a symmetric fiber functor to the symmetric tensor category [Formula: see text] of representations of the triangular Hopf algebra [Formula: see text]. Equivalently, we prove that there exists a unique finite group scheme [Formula: see text] in [Formula: see text] such that [Formula: see text] is symmetric tensor equivalent to [Formula: see text]. Finally, we compute the group [Formula: see text] of equivalence classes of twists for the group algebra [Formula: see text] of a finite abelian [Formula: see text]-group [Formula: see text] over an arbitrary field [Formula: see text] of characteristic [Formula: see text], and the Sweedler cohomology groups [Formula: see text], [Formula: see text], of the function algebra [Formula: see text] of [Formula: see text].

Constructing colimits by gluing vector bundles

As already observed by Gabriel, coherent sheaves on schemes obtained by gluing affine open subsets can be described by a simple gluing construction. An example due to Ferrand shows that this fails in general for pushouts along closed immersions, though the gluing construction still works for flat coherent sheaves.We show that by further restricting this gluing construction to vector bundles, we can construct pushouts along arbitrary morphisms (and more general colimits) of certain algebraic stacks called Adams stacks. The proof of this fact uses generalized Tannaka duality and a variant of Deligne's argument for the existence of fiber functors which works in arbitrary characteristic.We use this version of Deligne's existence theorem for fiber functors as a novel way of recognizing stacks which have atlases. It differs considerably from Artin's algebraicity results and their generalizations: rather than studying conditions on the functor of points which ensure the existence of an atlas, our theorem identifies conditions on the category of quasi-coherent sheaves of the stack which imply that an atlas exists.

Category of C-motives over finite fields

In this article we introduce and study a motivic category in the arithmetic of function fields, namely the category of motives over an algebraic closure L of a finite field with coefficients in a global function field over this finite field. It is semi-simple, non-neutral Tannakian and possesses all the expected fiber functors. This category generalizes the previous construction due to Anderson and is more relevant for applications to the theory of G-Shtukas, such as formulating the analog of the Langlands-Rapoport conjecture over function fields. We further develop the analogy with the category of motives over L with coefficients in Q for which the existence of the expected fiber functors depends on famous unproven conjectures.

Filtrations and Buildings

We construct and study a scheme theoretical version of the Tits vectorial building, relate it to filtrations on fiber functors, and use them to clarify various constructions pertaining to Bruhat-Tits buildings, for which we also provide a Tannakian description.

On symmetric fusion categories in positive characteristic

We propose a conjectural extension in the positive characteristic case of 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.

Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and geometric Satake equivalence

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.

Classifying module categories for generalized Temperley–Lieb–Jones ∗-2-categories

Generalized Temperley–Lieb–Jones (TLJ) 2-categories associated to weighted bidirected graphs were introduced in unpublished work of Morrison and Walker. We introduce unitary modules for these generalized TLJ 2-categories as strong ∗-pseudofunctors into the ∗-2-category of row-finite separable bigraded Hilbert spaces. We classify these modules up to ∗-equivalence in terms of weighted bi-directed fair and balanced graphs in the spirit of Yamagami’s classification of fiber functors on TLJ categories and DeCommer and Yamashita’s classification of unitary modules for [Formula: see text].

Coxeter categories and quantum groups

We define the notion of braided Coxeter category, which is informally a tensor category carrying compatible, commuting actions of a generalised braid group B_W and Artin's braid groups B_n on the tensor powers of its objects. The data which defines the action of B_W bears a formal similarity to the associativity constraints in a monoidal category, but is related to the coherence of a family of fiber functors. We show that the quantum Weyl group operators of a quantised Kac-Moody algebra U_h(g), together with the universal R-matrices of its Levi subalgebras, give rise to a braided Coxeter structure on integrable, category O-modules for U_h(g). By relying on the 2-categorical extension of Etingof-Kazhdan quantisation obtained in arXiv:1610.09744, we then prove that this structure can be transferred to integrable, category O-representations of g. These results are used in arXiv:1512.03041 to give a monodromic description of the quantum Weyl group operators of U_h(g) which extends the one obtained by the second author for a semisimple Lie algebra.

Virtual tangles and fiber functors

We define a category [Formula: see text] of tangles diagrams drawn on surfaces with boundaries. On the one hand, we show that there is a natural functor from the category of virtual tangles to [Formula: see text] which induces an equivalence of categories. On the other hand, we show that [Formula: see text] is universal among ribbon categories equipped with a strong monoidal functor to a symmetric monoidal category. This is a generalization of the Shum–Reshetikhin–Turaev theorem characterizing the category of ordinary tangles as the free ribbon category. This gives a straightforward proof that all quantum invariants of links extend to framed oriented virtual links. This also provides a clear explanation of the relation between virtual tangles and Etingof–Kazhdan formalism suggested by Bar-Natan. We prove a similar statement for virtual braids, and discuss the relation between our category and knotted trivalent graphs.

Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof

AbstractWe prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber functor to the category of supervector spaces. This proves Ostrik’s conjecture [25, Conjecture 1.3] in this case. Equivalently, we prove that there exists a unique finite supergroup scheme $\mathcal{G}$ over $k$ and a grouplike element $\epsilon \in k\mathcal{G}$ of order $\le 2$, whose action by conjugation on $\mathcal{G}$ coincides with the parity automorphism of $\mathcal{G}$, such that $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(\mathcal{G},\epsilon )$. In particular, when $\mathcal{C}$ is unipotent, the functor lands in $\textrm{Vec}$, so $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(U)$ for a unique finite unipotent group scheme $U$ over $k$. We apply our result and the results of [17] to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over $k$ (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field $k$ of characteristic $p>0$, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field $k$ of characteristic $p>0$. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper [4], and more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic $\ne 2$ is always a Serre subcategory.