Equivalence Of Categories Research Articles

Abacus proofs of Schur expansions of Macdonald polynomials

A famous problem in symmetric function theory is to find combinatorial formulas for Schur expansions of the Macdonald polynomials H ~ μ \tilde {H}_{\mu } . One such formula, valid for μ \mu satisfying μ 1 ≤ 3 \mu _1\leq 3 and μ 2 ≤ 2 \mu _2\leq 2 , involves Yamanouchi words weighted by Haglund’s statistics inv μ \operatorname {inv}_{\mu } and maj μ \operatorname {maj}_{\mu } . Previous proofs of this formula use the technical machinery of crystals and dual equivalence graphs. We give a new, elementary, and fully bijective proof of this formula based on the abacus model for antisymmetrized Macdonald polynomials. An extension to the Schur expansion of s ν H ~ μ s_{\nu }\tilde {H}_{\mu } is also provided.

Choice-free duality for orthocomplemented lattices by means of spectral spaces

The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander’s Subbase Theorem, which asserts that a topological space X is compact if every subbasic open cover of X admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem—whose proof depends upon Zorn’s Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces; choice-free in the sense that our representation avoids use of Alexander’s Subbase Theorem, along with its associated nonconstructive choice principles. We then introduce a new subclass of spectral spaces which we call upper Vietoris orthospaces in order to characterize up to homeomorphism (and isomorphism with respect to their orthospace reducts) the spectral spaces of proper lattice filters used in our representation. It is then shown how our constructions give rise to a choice-free dual equivalence of categories between the category of orthocomplemented lattices and the dual category of upper Vietoris orthospaces. Our duality combines Bezhanishvili and Holliday’s choice-free spectral space approach to Stone duality for Boolean algebras with Goldblatt and Bimbó’s choice-dependent orthospace approach to Stone duality for orthocomplemented lattices.

Local Systems on Diamonds and p-Adic Vector Bundles

Abstract We use Scholze’s framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper, we provide a detailed analysis of local systems on diamonds for the étale, pro-étale, and the $v$-topology and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $\mathbb {C}_p$, this enables us to prove a category equivalence between $\mathbb {C}_p$-local systems with integral models, and modules under the $v$-structure sheaf that modulo each $p^n$ can be trivialized on a proper cover. The flexibility of the $v$-topology together with a descent result on integral models of local systems allows us to prove that the trivializability condition in the module category may be checked on any normal proper cover. This result leads to an extension of the parallel transport theory by Deninger and the second author to vector bundles with numerically flat reduction on a proper normal cover. 2020 MSC: 14G45, 14G22, 11G25.

Broadband Sound Insulation and Dual Equivalent Negative Properties of Acoustic Metamaterial with Distributed Piezoelectric Resonators.

Aiming at the unsatisfactory sound transmission loss (STL) of thin-plate structures in the low-mid frequency range, this paper proposes an acoustic insulation metamaterial with distributed piezoelectric resonators. A complete acoustic prediction model is established based on the effective medium method and classical plate theory, and the correctness is verified by the STL simulation results of the corresponding acoustic-structure fully coupled finite-element model. Moreover, the intrinsic relationship between the dual equivalent negative properties and STLs is investigated to reveal the insulation mechanisms of this metamaterial. Then, the influence of the geometric and material parameters on the double equivalent negative characteristics is studied to explore the broadband STL for distributed multi-modal resonant energy-dissipation modes in the frequency band of interest. The results show that the two acoustic insulation crests correspond to the dual equivalent negative performances, and the sound insulation in the low-mid frequency range is improved by more than 5 dB compared with that of the substrate, even up to 44.49 dB.

Equivariant categories of symplectic surfaces and fixed loci of Bridgeland moduli spaces

Given an action of a finite group $G$ on the derived category of a smooth projective variety $X$ we relate the fixed loci of the induced $G$-action on moduli spaces of stable objects in $D^b(\mathrm{Coh}(X))$ with moduli spaces of stable objects in the equivariant category $D^b(\mathrm{Coh}(X))_G$. As an application we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence, and yields a general framework for describing fixed loci of symplectic group actions on moduli spaces of stable objects on symplectic surfaces.

A tale of two categories: Inductive groupoids and cross-connections

A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialized groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialized small category whose object set is a strict preorder and the morphisms admit a factorization property . A pair of ‘related’ normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad as categorical models of regular semigroups. We explore the inter-relationship between these seemingly different categorical structures and prove a direct category equivalence between the category of inductive groupoids and the category of cross-connections.

Equivariant approach to weighted projective curves

We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations.This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a description for the category of coherent sheaves over a hyperelliptic curve of arbitrary genus. Links to smooth projective curves of genus 2 and also to Arnold's exceptional unimodal singularities are also established.

Deformed dimensional reduction

Since its first use by Behrend, Bryan, and Szendr\H{o}i in the computation of motivic Donaldson-Thomas (DT) invariants of $\mathbb{A}_{\mathbb{C}}^3$, dimensional reduction has proved to be an important tool in motivic and cohomological DT theory. Inspired by a conjecture of Cazzaniga, Morrison, Pym, and Szendr\H{o}i on motivic DT invariants, work of Dobrovolska, Ginzburg, and Travkin on exponential sums, and work of Orlov and Hirano on equivalences of categories of singularities, we generalize the dimensional reduction theorem in motivic and cohomological DT theory and use it to prove versions of the Cazzaniga-Morrison-Pym-Szendr\H{o}i conjecture in these settings.

Endomorphism algebras and Hecke algebras for reductive p-adic groups

Let G be a reductive p-adic group and let Rep(G)s be a Bernstein block in the category of smooth complex G-representations. We investigate the structure of Rep(G)s, by analysing the algebra of G-endomorphisms of a progenerator Π of that category.We show that Rep(G)s is “almost” Morita equivalent with a (twisted) affine Hecke algebra. This statement is made precise in several ways, most importantly with a family of (twisted) graded algebras. It entails that, as far as finite length representations are concerned, Rep(G)s and EndG(Π)-Mod can be treated as the module category of a twisted affine Hecke algebra.We draw two major consequences. Firstly, we show that the equivalence of categories between Rep(G)s and EndG(Π)-Mod preserves temperedness of finite length representations. Secondly, we provide a classification of the irreducible representations in Rep(G)s, in terms of the complex torus and the finite group canonically associated to Rep(G)s. This proves a version of the ABPS conjecture and enables us to express the set of irreducible G-representations in terms of the supercuspidal representations of the Levi subgroups of G.Our methods are independent of the existence of types, and apply in complete generality.

Air pollution and executive incentive: Evidence from pay-performance sensitivity

Based on the China's non-state-owned listed corporates in 2014–2020, we adopt the real-time air quality index data published by the Ministry of Environmental Protection of China as the proxy of air pollution to examine how air pollution affects firm's CEO pay–performance sensitivity (PPS). The results of logistic regression show that air pollution is negatively correlated with the executives' PPS. We also find that industrial and regional characteristics is heterogeneous by exploring the interaction effect of CEO. In addition, our study indicates that the influence of air pollution on PPS is more significant in companies with improved performance and we provide a possible explanation of this based on the theory of resource category equivalence. The mechanism test shows that air pollution increases the firms' difficulty to motivate executives, it also destroys the effectiveness of compensation contracts and significantly reduces the PPS by increasing explicit and implicit incentive. Furthermore, we investigate the economic consequences of air pollution. Results show that air pollution would damages the firm value directly and also affect their PPS. Overall, our study reveals how air pollution affects executives' incentive, therefore provides policy support to developing countries to balance the relationship between economic development and environmental protection.

Group superschemes

We develop a general theory of algebraic group superschemes, which are not necessarily affine. Our key result is a category equivalence between those group superschemes and Harish-Chandra pairs, which generalizes the result known for affine algebraic group superschemes. Then we present the applications, including the Barsotti-Chevalley Theorem in the super context, and an explicit construction of the quotient superscheme G/H of an algebraic group superscheme G by a group super-subscheme H.

Coherent categorical structures for Lie bialgebras, Manin triples, classical r-matrices and pre-Lie algebras

Abstract The broadly applied notions of Lie bialgebras, Manin triples, classical r-matrices and 𝒪 {\mathcal{O}} -operators of Lie algebras owe their importance to the close relationships among them. Yet these notions and their correspondences are mostly understood as classes of objects and maps among the classes. To gain categorical insight, this paper introduces, for each of the classes, a notion of homomorphisms, uniformly called coherent homomorphisms, so that the classes of objects become categories and the maps among the classes become functors or category equivalences. For this purpose, we start with the notion of an endo Lie algebra, consisting of a Lie algebra equipped with a Lie algebra endomorphism. We then generalize the above classical notions for Lie algebras to endo Lie algebras. As a result, we obtain the notion of coherent endomorphisms for each of the classes, which then generalizes to the notion of coherent homomorphisms by a polarization process. The coherent homomorphisms are compatible with the correspondences among the various constructions, as well as with the category of pre-Lie algebras.

A Serre–Swan theorem for coisotropic algebras

Coisotropic algebras are used to formalize coisotropic reduction in Poisson geometry as well as in deformation quantization and find applications in various other fields as well. In this paper we prove a Serre-Swan Theorem relating the regular projective modules over the coisotropic algebra built out of a manifold $M$, a submanifold $C$ and an integrable smooth distribution $D \subseteq TC$ with vector bundles over this geometric situation and show an equivalence of categories for the case of a simple distribution.

Mukai duality via roofs of projective bundles

We investigate a construction providing pairs of Calabi–Yau varieties described as zero loci of pushforwards of a hyperplane section on a roof as described in Kanemitsu (Preprint, arXiv:1812.05392, 2018). We discuss the implications of such construction at the level of Hodge equivalence, derived equivalence and L ${\mathbb {L}}$ -equivalence. For the case of K3 surfaces, we provide alternative interpretations for the Fourier–Mukai duality in the family of K3 surfaces of degree 12 of (New trends in algebraic geometry (Warwick, 1996), Lond. Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 311–326). In all these constructions, the derived equivalence lifts to an equivalence of matrix factorizations categories.

Twist Structures and Nelson Conuclei

Motivated by Kalman residuated lattices, Nelson residuated lattices and Nelson paraconsistent residuated lattices, we provide a natural common generalization of them. Nelson conucleus algebras unify these examples and further extend them to the non-commutative setting. We study their structure, establish a representation theorem for them in terms of twist structures and conuclei that results in a categorical adjunction, and explore situations where the representation is actually an isomorphism. In the latter case, the adjunction is elevated to a categorical equivalence. By applying this representation to the original motivating special cases we bring to the surface their underlying similarities.

Stone dualities from opfibrations

Stone dualities are dual equivalences between certain categories of algebras and those of topological spaces. A Stone duality is often derived from a dual adjunction between such categories by cutting down unnecessary objects. This dual adjunction is called the fundamental adjunction of the duality, but building it often requires concrete topological arguments. The aim of this paper is to construct fundamental adjunctions generically using (op)fibered category theory. This paper defines an abstract notion of formal spaces (including ordinary topological spaces as the leading example), and gives a construction of a fundamental adjunction between the category of algebras and the category of corresponding formal spaces. Moreover, prove an Adjoint Lifting Theorem in the setting of near 2-fibrations, and construct the fundamental adjunction of Priestley duality.

Nilpotency and the Hamiltonian property for cancellative residuated lattices

This paper studies nilpotent and Hamiltonian cancellative residuated lattices and their relationship with nilpotent and Hamiltonian lattice-ordered groups. In particular, results about lattice-ordered groups are extended to the domain of residuated lattices. The two key ingredients that underlie the considerations of this paper are the categorical equivalence between Ore residuated lattices and lattice-ordered groups endowed with a suitable modal operator; and Malcev’s description of nilpotent groups of a given nilpotency class c in terms of a semigroup equation.

Dualization between supersymmetric self-dual and topologically massive models coupled to matter in four-dimensional superspace

Using the superfield formalism and the master action approach, we prove, both at the classical and quantum levels, the dual equivalence between four-dimensional supersymmetric self-dual and topologically massive models coupled to dynamical matter.

Skew key polynomials and a generalized Littlewood–Richardson rule

Young’s lattice is a partial order on integer partitions whose saturated chains correspond to standard Young tableaux, one type of combinatorial object that generates the Schur basis for symmetric functions. Generalizing Young’s lattice, we introduce a new partial order on weak compositions that we call the key poset. Saturated chains in this poset correspond to standard key tableaux, the combinatorial objects that generate the key polynomials, a nonsymmetric polynomial generalization of the Schur basis. Generalizing skew Schur functions, we define skew key polynomials in terms of this new poset. Using weak dual equivalence, we give a nonnegative weak composition Littlewood–Richardson rule for the key expansion of skew key polynomials, generalizing the flagged Littlewood–Richardson rule of Reiner and Shimozono.

A description of monodromic mixed Hodge modules

Abstract For a smooth algebraic variety X, a monodromic D-module on X × ℂ {X\times\mathbb{C}} is decomposed into a direct sum of some D-modules on X. We show that the Hodge filtration of a mixed Hodge module on X × ℂ {X\times\mathbb{C}} whose underlying D-module is monodromic is also decomposed. Moreover, we show that there is an equivalence of categories between the category of monodromic mixed Hodge modules on X × ℂ {X\times\mathbb{C}} and the category of “gluing data”. As an application, we endow the Fourier–Laplace transformation of the underlying D-module of a monodromic mixed Hodge module with a mixed Hodge module structure.