We construct a separable Frobenius monoidal functor from $\mathcal{Z}\big(\mathsf{Vect}_H^{ω|_H}\big)$ to $\mathcal{Z}\big(\mathsf{Vect}_G^ω\big)$ for any subgroup $H$ of $G$ which preserves braiding and ribbon structure. As an application, we classify rigid Frobenius algebras in $\mathcal{Z}\big(\mathsf{Vect}_G^ω\big)$, recovering the classification of étale algebras in these categories by Davydov-Simmons [J. Algebra 471 (2017), 149-175, arXiv:1603.04650] and generalizing their classification to algebraically closed fields of arbitrary characteristic. Categories of local modules over such algebras are modular tensor categories by results of Kirillov-Ostrik [Adv. Math. 171 (2002), 183-227, arXiv:math.QA/0101219] in the semisimple case and Laugwitz-Walton [Comm. Math. Phys., to appear, arXiv:2202.08644] in the general case.
In this work, for a finite group $G$ and a 4-cocycle $ω\in Z^4(G, \mathbf{k}^\times)$, we compute explicitly the center of the monoidal 2-category $\operatorname{2Vec}_G^ω$ of $ω$-twisted $G$-graded 1-categories of finite dimensional $\mathbf{k}$-vector spaces. This center gives a precise mathematical description of the topological defects in the associated 3+1D Dijkgraaf-Witten TQFT. We prove that this center is a braided monoidal 2-category with a trivial sylleptic center.
We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.
Let $U:\mathcal{C}\rightarrow\mathcal{D}$ be a strong monoidal functor between abelian monoidal categories admitting a right adjoint $R$, such that $R$ is exact, faithful and the adjunction $U\dashv R$ is coHopf. Building on the work of Balan, we show that $R$ is separable (resp., special) Frobenius monoidal if and only if $R(\mathbb{1}_{\mathcal{D}})$ is a separable (resp., special) Frobenius algebra in $\mathcal{C}$. If further, $\mathcal{C},\mathcal{D}$ are pivotal (resp., ribbon) categories and $U$ is a pivotal (resp., braided pivotal) functor, then $R$ is a pivotal (resp., ribbon) functor…
Separable Frobenius monoidal functors were defined and studied under that name by Szlachanyi and by Day and Pastro, and in a more general context by Cockett and Seely. Our purpose here is to develop their theory in a very precise sense. We determine what kinds of equations in monoidal categories they preserve. For example we show they preserve lax (meaning not necessarily invertible) Yang-Baxter operators, weak Yang-Baxter operators in the sense of Alonso Alvarez et al., and (in the braided case) weak bimonoids in the sense of Pastro and Street. In fact, we characterize which monoidal expressi…
We study monoidal comonads on a naturally Frobenius map-monoidale $M$ in a monoidal bicategory $\mathcal M$. We regard them as bimonoids in the duoidal hom-category $\mathcal M(M,M)$, and generalize to that setting various conditions distinguishing classical Hopf algebras among bialgebras; in particular, we define a notion of antipode in that context. Assuming the existence of certain conservative functors and the splitting of idempotent 2-cells in $\mathcal M$, we show all these Hopf-like conditions to be equivalent. Our results imply in particular several equivalent characterizations of Hopf…
We characterize Frobenius and separable monoidal algebra extensions $i: R\ra S$ in terms given by $R$ and $S$. For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if $S$ is a Frobenius, respectively separable, algebra in the category of bimodules over $R$. In the case when $R$ is separable we show that the extension is separable if and only if $S$ is a separable algebra. Similarly, in the case when $R$ is Frobenius and separable in a sovereign monoidal category we show that the extension is Frobenius if and only if $S$ is a Frobeniu…
Let $U$ be a strong monoidal functor between monoidal categories. If it has both a left adjoint $L$ and a right adjoint $R$, we show that the pair $(R,L)$ is a linearly distributive functor and $(U,U)\dashv (R,L)$ is a linearly distributive adjunction, if and only if $L\dashv U$ is a Hopf adjunction and $U\dashv R$ is a coHopf adjunction. We give sufficient conditions for a strong monoidal $U$ which is part of a (left) Hopf adjunction $L\dashv U$, to have as right adjoint a twisted version of the left adjoint $L$. In particular, the resulting adjunction will be (left) coHopf. One step further,…
We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a $k$-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a $k$-linear semisimple pivotal monoidal category -- where both notions are defined --, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.
In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. In particular, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction in some 2-category. Our theorem is proved by extending the theory of adjoint monads to the context of…
We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs-Stigner. Thes…
We show that a left-rigid monoidal C^*-category with irreducible monoidal unit is also a sovereign and spherical category. Defining a Frobenius-Schur type indicator we obtain selection rules for the fusion coefficients of irreducible objects. As a main result we prove S_4-invariance of 6j-symbols in such a category.
We consider Frobenius algebras and their bimodules in certain abelian monoidal categories. In particular we study the Picard group of the category of bimodules over a Frobenius algebra, i.e. the group of isomorphism classes of invertible bimodules. The Rosenberg-Zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the Picard group, which however is typically not surjective. We investigate under which conditions there exists a Morita equivalent Frobenius algebra for which the corresponding homomorphism is surjective. One motivation for our considerations is the …
We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R \otimes R.
The monoidal version of classical Morita theory is a theory of bialgebroids. To make this explicit we construct a bicategory the objects of which are the bialgebroids and in which equivalence of objects means that the corresponding module categories are monoidally equivalent. The module categories of bialgebroids and of Frobenius Hopf algebroids are characterized by the existence of strong comonoid progenerators and strong Frobenius progenerators, respectively.
In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius-Schur exponents. We classify spherical fusion categories of Frobenius-Schur exponent 2 up to monoidal equivalence. We also classify modular categories of Frobenius-Schur exponent 2 up to braided monoidal equivalence. It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius-Schur exponent 2. This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.
In this paper, we introduce the notion of the pivotal cover $\mathcal{C}^{\mathsf{piv}}$ of a left rigid monoidal category $\mathcal{C}$ to develop a theoretical foundation for the theory of Frobenius-Schur (FS) indicators in "non-pivotal" settings. For an object $\mathbf{V} \in \mathcal{C}^{\mathsf{piv}}$, the $(n, r)$-th FS indicator $ν_{n, r}(\mathbf{V})$ is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators. Based on our framework, we also study the FS indicators of t…
We associate a diagrammatic monoidal category $\mathcal{H}\textit{eis}_k(A;z,t)$, which we call the quantum Frobenius Heisenberg category, to a symmetric Frobenius superalgebra $A$, a central charge $k \in \mathbb{Z}$, and invertible parameters $z,t$ in some ground ring. When $A$ is trivial, i.e. it equals the ground ring, these categories recover the quantum Heisenberg categories introduced in our previous work, and when the central charge $k$ is zero they yield generalizations of the affine HOMFLY-PT skein category. By exploiting some natural categorical actions of $\mathcal{H}\textit{eis}_k…
We study integrals of Hopf monoids in duoidal endohom categories of naturally Frobenius map monoidales in monoidal bicategories. We prove two Maschke type theorems, relating the separability of the underlying monoid and comonoid, respectively, to the existence of normalized integrals. It covers the examples provided by Hopf monoids in braided monoidal categories, weak Hopf algebras, Hopf algebroids over central base algebras, Hopf monads on autonomous monoidal categories and Hopf categories.
Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the category of 2-dimensional cobordisms can be described as the free symmetric monoidal category on a commutative Frobenius algebra, 2Thick is shown to be the free monoidal category on a noncommutative Frobenius algebra, hence justifying this choice of data in the Moore-Segal axioms. Our…
We introduce a variant on the graphical calculus of Cockett and Seely for monoidal functors and illustrate it with a discussion of Tannaka reconstruction, some of which is known and some of which is new. The new portion is: given a separable Frobenius functor F: A --> B from a monoidal category A to a suitably complete or cocomplete braided autonomous category B, the usual formula for Tannaka reconstruction gives a weak bialgebra in B; if, moreover, A is autonomous, this weak bialgebra is in fact a weak Hopf algebra.
We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra $H$. If $H$ is a group Hopf algebra, we study a more general Frobenius type property and uncover the structure of graded Frobenius algebras. Graded symmetric algebras are also investigated.
Dijkgraaf, Pasquier and Roche introduced twisted quantum doubles of a finite group in the context of conformal field theory. We study equivalences that arise among the braided monoidal categories associated to these quantum doubles, especially in the commutative case. This involves a close study of the cohomology of various complexes introduced by Eilenberg-MacLane. We show among other things that an equivalence of braided monoidal categories is the same as gauge equivalence of the corresponding quantum doubles, and that an invariant for an equivalence class is given by a metabolic quadratic s…
Two-dimensional conformal field theory (CFT) can be defined through its correlation functions. These must satisfy certain consistency conditions which arise from the cutting of world sheets along circles or intervals. The construction of a (rational) CFT can be divided into two steps, of which one is complex-analytic and one purely algebraic. We realise the algebraic part of the construction with the help of three-dimensional topological field theory and show that any symmetric special Frobenius algebra in the appropriate braided monoidal category gives rise to a solution. A special class of e…
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…
A modular object in a symmetric monoidal bicategory is a Frobenius algebra object whose product and coproduct are biadjoint, equipped with a braided structure and a compatible twist, satisfying rigidity, ribbon, pivotality, and modularity conditions. We prove that the oriented 3-dimensional bordism bicategory of 1-, 2-, and 3-manifolds is the free symmetric monoidal bicategory on a single anomaly-free modular object.
For any finite-dimensional factorizable ribbon Hopf algebra H and any ribbon automorphism omega of H, we establish the existence of the following structure: an H-bimodule F_omega and a bimodule morphism Z_omega from Lyubashenko's Hopf algebra object K for the bimodule category to F_omega. This morphism is invariant under the natural action of the mapping class group of the one-punctured torus on the space of bimodule morphisms from K to F_omega. We further show that the bimodule F_omega can be endowed with a natural structure of a commutative symmetric Frobenius algebra in the monoidal categor…
For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion is formulaic while the latter is categorical. We generalize this result to the setting of weak bialgebras $H$. The category $\mathcal{M}^H$ admits a monoidal structure by work of Nill and Böhm-Caenepeel-Janssen, but the algebras in $\mathcal{M}^H$ are not canonically $\Bbbk$-algebras. Nevertheless, we prove that there is an isomorphism between the category o…
We consider a pivotal monoidal functor whose domain is a modular tensor category (MTC). We show that the trace of such a functor naturally extends to a representation of the corresponding tube category. As irreducible representations of the tube category are indexed by pairs of simple objects in the underlying MTC, the simple multiplicities of this representation form a candidate modular invariant matrix. In general, this matrix will not be modular invariant, however it will always commute with the T-matrix. Furthermore, under certain additional conditions on the original functor, it is shown …
We compute higher Frobenius-Schur indicators of pq-dimensional pointed Hopf algebras in characteristic p through their associated graded Hopf algebras. These indicators are gauge invariants for the monoidal categories of representations of these algebras.
We show that every modular category is equivalent as an additive ribbon category to the category of finite-dimensional comodules of a Weak Hopf Algebra. This Weak Hopf Algebra is finite-dimensional, split cosemisimple, weakly cofactorizable, coribbon and has trivially intersecting base algebras. In order to arrive at this characterization of modular categories, we develop a generalization of Tannaka-Krein reconstruction to the long version of the canonical forgetful functor which is lax and oplax monoidal, but not in general strong monoidal, thereby avoiding all the difficulties related to non…
We associate a monoidal category $\mathcal{H}_B$, defined in terms of planar diagrams, to any graded Frobenius superalgebra $B$. This category acts naturally on modules over the wreath product algebras associated to $B$. To $B$ we also associate a (quantum) lattice Heisenberg algebra $\mathfrak{h}_B$. We show that, provided $B$ is not concentrated in degree zero, the Grothendieck group of $\mathcal{H}_B$ is isomorphic, as an algebra, to $\mathfrak{h}_B$. For specific choices of Frobenius algebra $B$, we recover existing results, including those of Khovanov and Cautis--Licata. We also prove tha…
We study Galois extensions Coinv(M)<M for M an H-comodule algebra and H a Frobenius Hopf algebroid. We obtain generalizations of various theorems in Hopf-Galois theory by Kreimer-Takeuchi, Doi-Takeuchi and Cohen-Fischman-Montgomery. An algebra extension is Galois precisely if it is balanced, depth 2, and Frobenius. Then we show that Yetter-Drinfeld categories over H are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by the Brzezinski-Militaru Theorem. Contravariant "fiber functors" are used to prove an analogue of Ulbrich's Theorem and t…
We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed TQFTs. These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras. A knowledgeable Frobenius algebra (A,C,i,i^*) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an …
Control theory uses "signal-flow diagrams" to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted. These diagrams can be seen as string diagrams for the symmetric monoidal category FinVect_k of finite-dimensional vector spaces over the field of rational functions k = R(s), where the variable s acts as differentiation and the monoidal structure is direct sum rather than the usual tensor product of vector spaces. For any field k we give a presentation of FinVect_k in terms of the generators used in signal …
Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable H, our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds.
We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. Our focus is the Hecke category of Borel-equivariant D-modules on the flag variety of a complex reductive group G (equivalently, the category of Harish Chandra bimodules of trivial central character) and its monodromic variant. The Hecke category is a categorified analogue of the finite Hecke algebra, which is a finite-dimensional semi-simple symmetric Frobenius algebra. We establish parallel properties of the Hecke category, showing it is a two-dualizable Calabi-Yau …
Let $\mathcal{C}$ be a finite tensor category with simple unit object, let $\mathcal{Z}(\mathcal{C})$ denote its monoidal center, and let $L$ and $R$ be a left adjoint and a right adjoint of the forgetful functor $U: \mathcal{Z}(\mathcal{C}) \to \mathcal{C}$. We show that the following conditions are equivalent: (1) $\mathcal{C}$ is unimodular, (2) $U$ is a Frobenius functor, (3) $L$ preserves the duality, (4) $R$ preserves the duality, (5) $L(1)$ is self-dual, and (6) $R(1)$ is self-dual, where $1 \in \mathcal{C}$ is the unit object. We also give some other equivalent conditions. As an applic…
