Equivalence Of Categories Research Articles

Centres, trace functors, and cyclic cohomology

We study the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with their bimodule category centres over the respective opposite categories and a corresponding categorical equivalence to anti Yetter-Drinfel’d contramodules and anti Yetter-Drinfel’d modules, respectively. This is directly connected to the existence of a trace functor on the monoidal categories of modules and comodules in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology.

Complemented MacNeille completions and algebras of fractions

We introduce (ℓ-)bimonoids as ordered algebras consisting of two compatible monoidal structures on a partially ordered (lattice-ordered) set. Bimonoids form an appropriate framework for the study of a general notion of complementation, which subsumes both Boolean complements in bounded distributive lattices and multiplicative inverses in monoids. The central question of the paper is whether and how bimonoids can be embedded into complemented bimonoids, generalizing the embedding of cancellative commutative monoids into their groups of fractions and of bounded distributive lattices into their free Boolean extensions. We prove that each commutative (ℓ-)bimonoid embeds into a complete complemented commutative ℓ-bimonoid in a doubly dense way reminiscent of the Dedekind–MacNeille completion. Moreover, this complemented completion, which is term equivalent to a commutative involutive residuated lattice, sometimes contains a tighter complemented envelope analogous to the group of fractions. In the case of cancellative commutative monoids this algebra of fractions is precisely the familiar group of fractions, while in the case of Brouwerian (Heyting) algebras it is a (bounded) idempotent involutive commutative residuated lattice. This construction of the algebra of fractions in fact yields a categorical equivalence between varieties of integral and of involutive residuated structures which subsumes as special cases the known equivalences between Abelian ℓ-groups and their negative cones, and between Sugihara monoids and their negative cones.

A Tour of p-Permutation Modules and Related Classes of Modules

This survey provides an overview of numerous results on p-permutation modules and the closely related classes of endo-trivial, endo-permutation and endo-p-permutation modules. These classes of modules play an important role in the representation theory of finite groups. For example, they are important building blocks used to understand and parametrise several kinds of categorical equivalences between blocks of finite group algebras. For this reason, there has been, since the late 1990’s, much interest in classifying such modules. The aim of this manuscript is to review classical results as well as all the major recent advances in the area. The first part of this survey serves as an introduction to the topic for non-experts in modular representation theory of finite groups, outlining proof ideas of the most important results at the foundations of the theory. Simultaneously, the connections between the aforementioned classes of modules are emphasised. In this respect, results, which are dispersed in the literature, are brought together, and emphasis is put on common properties and the role played by the p-permutation modules throughout the theory. Finally, in the last part of the manuscript, lifting results from positive characteristic to characteristic zero are collected and their proofs sketched.

Planar Algebras in Braided Tensor Categories

We generalize Jones’ planar algebras by internalising the notion to a pivotal braided tensor category C \mathcal {C} . To formulate the notion, the planar tangles are now equipped with additional ‘anchor lines’ which connect the inner circles to the outer circle. We call the resulting notion an anchored planar algebra. If we restrict to the case when C \mathcal {C} is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in C \mathcal {C} and pivotal module tensor categories over C \mathcal {C} equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras.

Automatic Split-Generation for the Fukaya Category

Abstract We prove a structural result in mirror symmetry for projective Calabi–Yau (CY) manifolds. Let $X$ be a connected symplectic CY manifold, whose Fukaya category $\EuScript {F}(X)$ is defined over some suitable Novikov field ${\mathbb {K}}$; its mirror is assumed to be some smooth projective ${\mathbb {K}}$-variety $Y$ that is “maximally degenerating”. Suppose that some split-generating subcategory of (a $\textsf {dg}$ enhancement of) $D^bCoh( Y)$ embeds into $\EuScript {F}(X)$: we call this hypothesis “core homological mirror symmetry”. We prove that the embedding extends to an equivalence of categories, $D^{b}Coh(Y) \simeq D^{\pi }( \EuScript {F}(X))$, using Abouzaid’s split-generation criterion. The results only depend on certain formal properties of the Fukaya category, which have been established in certain cases but not in complete generality. The appendix, which can be read on its own, proves a result about Hochschild cohomology for schemes: the compatibility of the global Hochschild–Kostant–Rosenberg isomorphism with Kodaira–Spencer deformation theory.

Queer dual equivalence graphs

We introduce a new paradigm for proving the Schur $P$-positivity. Generalizing dual equivalence, we give an axiomatic definition for a family of involutions on a set of objects to be a queer dual equivalence, and we prove whenever such a family exists, the fundamental quasisymmetric generating function is Schur $P$-positive. In contrast with shifted dual equivalence, the queer dual equivalence involutions restrict to a dual equivalence when the queer involution is omitted. We highlight the difference between these two generalization with a new application to the product of Schur $P$-functions.

On the derived Lusztig correspondence

Abstract Let G be a connected reductive group, T a maximal torus of G, N the normalizer of T and $W=N/T$ the Weyl group of G. Let ${\mathfrak {g}}$ and ${\mathfrak {t}}$ be the Lie algebras of G and T. The affine variety $\mathfrak {car}={\mathfrak {t}}/\!/W$ of semisimple G-orbits of ${\mathfrak {g}}$ has a natural stratification $$ \begin{align*} \mathfrak{car}=\coprod_{\lambda}\mathfrak{car}_{\lambda} \end{align*} $$ indexed by the set of G-conjugacy classes of Levi subgroups: the open stratum is the set of regular semisimple orbits and the closed stratum is the set of central orbits. In [17], Rider considered the triangulated subcategory $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])^{\mathrm {Spr}}$ of $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])$ generated by the direct summand of the Springer sheaf. She proved that it is equivalent to the derived category of finitely generated dg modules over the smash product algebra ${\overline {\mathbb {Q}}_{\ell }}[W]\# H^{\bullet }_G(G/B)$ where $H^{\bullet }_G(G/B)$ is the G-equivariant cohomology of the flag variety. Notice that the later derived category is $D_{\mathrm {c}}^{\mathrm {b}}(\mathrm {B}(N))$ where $\mathrm {B}(N)=[\mathrm {Spec}(k)/N]$ is the classifying stack of N-torsors. The aim of this paper is to understand geometrically and generalise Rider’s equivalence of categories: For each $\lambda $ we construct a cohomological correspondence inducing an equivalence of categories between $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {t}}_{\lambda }/N])$ and $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\lambda }/G])^{\mathrm {Spr}}$ .

Cocycle deformations for weak Hopf algebras

In this paper we introduce a theory of multiplication alteration by two-cocycles for weak Hopf algebras. We show that, just like it happens for Hopf algebras, if H a weak Hopf algebra and H? its weak Hopf algebra deformation by a 2-cocycle ?, there is a braided monoidal category equivalence between the categories of left-right Yetter-Drinfel?d modules HYDH and H?YDH?. As a consequence we get in this context that the category Rep(D(H)) of left modules over the Drinfel?d double D(H) can be identified with the category Rep(D(H?)) of left modules over the Drinfel?d double D(H?).

Comparaison des nerfs n–catégoriques

Our aim is to compare three nerve functors for strict $n$-categories: the Street nerve, the cellular nerve and the multi-simplicial nerve. We show that these three functors are equivalent in some appropriate sense. In particular, the classes of $n$-categorical weak equivalences that they define coincide: they are the Thomason equivalences. We give two applications of this result: the first one states that a Dyer-Kan-type equivalence for Thomason equivalences is a Thomason equivalence; the second one, fundamental, is the stability of the class of Thomason equivalences under the dualities of the category of strict $n$-categories.

Equivariant Morita theory for graded tensor categories

We extend categorical Morita equivalence to finite tensor categories gra\-ded by a finite group $G$. We show that two such categories are graded Morita equivalent if and only if their equivariant Drinfeld centers are equivalent as braided $G$-crossed tensor categories.

Simple modules for quiver Hecke algebras and the Robinson–Schensted–Knuth correspondence

We formalize some known categorical equivalences to give a rigorous treatment of smooth representations of p-adic general linear groups, as ungraded modules over quiver Hecke algebras of type A. Graded variants of RSK-standard modules are constructed for quiver Hecke algebras. Exporting recent results from the p-adic setting, we describe an effective method for construction and classification of all simple modules as quotients of modules induced from maximal homogenous data. It is established that the products involved in the RSK construction fit the Kashiwara-Kim notion of normal sequences of real modules. We deduce that RSK-standard modules have simple heads, devise a formula for the shift of grading between RSK-standard and simple self-dual modules, and establish properties of their decomposition matrix, thus confirming expectations for p-adic groups raised in a work of the author with Lapid. Subsequent work will exhibit how the presently introduced RSK construction generalizes the much-studied Specht construction, when inflated from cyclotomic quotient algebras.

McKay correspondence and orbifold equivalence

We prove that a pair of singularities related by a transformation arising from the McKay correspondence are orbifold equivalent. From this we deduce a McKay type category equivalence for the matrix factorization categories.

Superized Troesch complexes and cohomology for strict polynomial superfunctors

We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of the (super)symmetric power functors . We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors. By an equivalence of categories, this also provides cohomology calculations in the category of left modules over Schur superalgebras.

Morita equivalence for graded rings

The classical Morita Theorem for rings established the equivalence of three statements involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. In our main result, we establish the equivalence of analogous statements involving graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings.

The Categorical Equivalence Between Domains and Interpolative Generalized Closure Spaces

The Categorical Equivalence Between Domains and Interpolative Generalized Closure Spaces

Perverse sheaves on $\mathbf{C}^{2}$ without vanishing cycles at the origin along a general plane curve with singularities

Generalizing MacPherson-Vilonen’s method [2] to arbitrary plane curve singularities, we provide a classification of perverse sheaves on the neighborhood of the origin in the complex plane, which are adapted to a germ of a complex analytic plane curve. We rely on the presentation of the fundamental group of the complement of the curve as obtained by Neto and Silva [5]. The main result is an equivalence of categories between the category of perverse sheaves on $ \mathbf{C}^{2}$ stratified with respect to a singular plane curve and the category of $n$-tuples of finite dimensional vector spaces and linear maps satisfying a finite number of suitable relations. As an application, we classify perverse sheaves with no vanishing cycles at the origin for a special case.

Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds

It was recently shown (by the second author and D\'{i}az Garc\'{i}a, Krutov, Somberg, and Strung) that every relative line module over an irreducible quantum flag manifold $\mathcal{O}_q(G/L_S)$ admits a unique $\mathcal{O}_q(G)$-covariant connection with respect to the Heckenberger-Kolb differential calculus $\Omega^1_q(G/L_S)$. In this paper we show that these connections are bimodule connections with an invertible associated bimodule map. This is proved by applying general results of Beggs and Majid, on principal connections for quantum principal bundles, to the quantum principal bundle presentation of the Heckenberger-Kolb calculi recently constructed by the authors and D\'{i}az Garc\'{i}a. Explicit presentations of the associated bimodule maps are given first in terms of generalised quantum determinants, then in terms of the FRT presentation of the algebra $\mathcal{O}_q(G)$, and finally in terms of Takeuchi's categorical equivalence for relative Hopf modules.

Invariant differential operators and the generalized symmetric group

In this paper we study the decomposition of the direct image of π+(OX) the polynomial ring OX as a D-module, under the map π: spec OX →spec OXG(r,n), where OXG(r,n) is the ring of invariant polynomial under the action of the wreath product G(r, p):= Z/rZ ~Sn. We first describe the generators of the simple components of π+(OX) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants, differential operators, on the higher Specht polynomials

Frobenius functors, stable equivalences and K-theory of Gorenstein projective modules

Owing to the difference in K-theory, an example by Dugger and Shipley implies that the equivalence of stable categories of Gorenstein projective modules should not be a Quillen equivalence. We give a sufficient and necessary condition for the Frobenius pair of faithful functors between two abelian categories to be a Quillen equivalence, which is also equivalent to that the Frobenius functors induce mutually inverse equivalences between stable categories of Gorenstein projective objects.We show that the category of Gorenstein projective objects is a Waldhausen category, then Gorenstein K-groups are introduced and characterized. As applications, we show that stable equivalences of Morita type preserve Gorenstein K-groups, CM-finiteness and CM-freeness. Two specific examples of path algebras are presented to illustrate the results, for which the Gorenstein K0 and K1-groups are calculated.

A dual equivalence for cofinal quantum B-algebras

This paper introduces weak logical algebraic domains and a special class of quantum B-algebras, namely, cofinal quantum B-algebras. It proves that there is a dual equivalence between the category of cofinal quantum B-algebras and that of weak logical algebraic domains. In particular, the category of residuated posets is shown to be dual equivalent to that of logical algebraic domains. This correspondence generalizes the classical duality of posets and algebraic domains.