Search

The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $Rep_R(G)$, the category of smooth representations of a $p$-adic group $G$ with coefficients in $R$. In particular, they were used to construct level 0 decompositions when $R=\overline{\mathbb{Z}}_{\ell}$, $\ell \neq p$, by Dat for $GL_n$ and the author for a more general group. Wang proved in the case of $GL_n$ that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of $GL_n$ and a unipotent block of another group. In this paper, we generalize Wang's equivalence of category to a connected reductive group on a non-archimedean local field., 17 pages, in English

In this two-part paper, we introduce a $$p$$ -adic analytic analogue of Backelin and Kremnizer’s construction of the quantum flag variety of a semisimple algebraic group, when $$q$$ is not a root of unity and $$\vert q-1\vert

Let X be a complete toric variety equipped with the action of a torus T, and G a reductive algebraic group, defined over an algebraically closed field K. We introduce the notion of a compatible ∑-filtered algebra associated to X, generalizing the notion of a compatible ∑-filtered vector space due to Klyachko, where ∑ denotes the fan of X. We combine Klyachko's classification of T-equivariant vector bundles on X with Nori's Tannakian approach to principal G-bundles, to give an equivalence of categories between T-equivariant principal G-bundles on X and certain compatible ∑-filtered algebras associated to X, when the characteristic of K is 0.

Gelfand Duality establishes a duality, i.e. a contravariant equivalence, between convenient categories of topological spaces and certain algebras of continuous functions. There has been much research in the case of commutative algebras, whereas the noncommutative case is still being developed. In the classical commutative case, Gelfand Duality gives a duality between compact Hausdorff spaces and commutative C*-Algebras and since then has been extended to many more classes of spaces. The noncommutative case includes Noncommutative Measure Theory, Noncommutative K-Theory and many other subjects incorporated in the field of Noncommutative Geometry. This thesis presents an overview of the aforementioned subjects, concentrating in the commutative case and commenting on topics of current research in the direction of Noncommutative Geometry. Η Δυϊκότητα Gelfand αφορά μια σχέση δυϊκότητας, δηλαδή μια αντισταθμιστική ισοδυναμία μεταξύ κατηγοριών, με συγκεκριμένες καλές ιδιότητες, τοπολογικών χώρων και αλγεβρών συνεχών συναρτήσεων. Η περίπτωση των μεταθετικών αλγεβρών έχει μελετηθεί διεξοδικά, ενώ αυτή των μη μεταθετικών είναι ακόμα υπό εξέλιξη. Στην κλασική μεταθετική περίπτωση, η Δυϊκότητα Gelfand μας παρέχει μια δυϊκότητα μεταξύ συμπαγών χώρων Hausdorff και μεταθετικών C* αλγεβρών και, περαιτέρω, έχει επεκταθεί σε πολλές άλλες κλάσεις τοπολογικών χώρων. Η μη μεταθετική περίπτωση περιλαμβάνει τη Μη Μεταθετική Θεωρία Μέτρου, τη Μη Μεταθετική K-θεωρία αλλά και αρκετά ακόμη θέματα, τα οποία εντάσσονται στον ευρύτερο κλάδο της Μη Μεταθετικής Γεωμετρίας. Η παρούσα διατριβή αποτελεί μια επισκόπηση των θεμάτων που προαναφέραμε, δίνοντας έμφαση στην μεταθετική περίπτωση και κάνοντας αναφορά σε θέματα που αποτελούν αντικείμενο ενεργούς έρευνας, προς την κατεύθυνση της Μη Μεταθετικής Γεωμετρίας. 103 σ.

Lie algebras and groups equipped with a multiplication $$\mu $$μ satisfying some compatibility properties are studied. These structures are called symmetric Lie $$\mu $$μ-algebras and symmetric $$\mu $$μ-groups respectively. An equivalence of categories between symmetric Lie $$\mu $$μ-algebras and symmetric Leibniz algebras is established when 2 is invertible in the base ring. The second main result of the paper is an equivalence of categories between simply connected symmetric Lie $$\mu $$μ-groups and finite dimensional symmetric Leibniz algebras.

We prove that intermediate extensions of perverse constructible F p -sheaves commute with smooth pullbacks for embeddable schemes over a field of characteristic p. Along the way we also prove that the equivalence of categories of Cartier crystals with unit R [ F ] -modules commutes with f ! for a smooth morphism f : X → Y of embeddable schemes.

In the previous chapter, we showed that Brandt matrices for an order in a definite quaternion algebra B contain a wealth of arithmetic. In the special case where \({{\,\mathrm{disc}\,}}B=p\) is prime, there is a further beautiful connection between Brandt matrices and the theory of supersingular elliptic curves, arising from the following important result: there is an equivalence of categories between supersingular elliptic curves over \(\overline{\mathbb{F }}_p\) and right ideals in a (fixed) maximal order \(\mathcal {O}\subset B\). We pursue this important connection in this chapter for the reader who has a bit more background in algebraic curves.

For a connected reductive group $G$ over a finite field, we study automorphic vector bundles on the stack of $G$-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski--Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of $G$-zips and a category of admissible modules with actions of a zero-dimensional algebraic subgroup a Levi subgroup and monodromy operators., Comment: 33 pages

An equivalence between ep-groupoids is the sc-smooth version of an equivalence of categories. In this chapter we shall study notions which are invariant under equivalences and introduce the notion of generalized maps.

From every pair of adjoint functors it is possible to produce a (possibly trivial) equivalence of categories by restricting to the subcategories where the unit and counit are isomorphisms. If we do this for the adjunction between effect algebras and abstract convex sets, we get the surprising result that the equivalent subcategories consist of reflexive order-unit spaces and reflexive base-norm spaces, respectively. These are the convex sets that can occur as state spaces in generalized probabilistic theories satisfying both the no-restriction hypothesis and its dual. The linearity of the morphisms is automatic. If we add a compact topology to either the states or the effects, we can obtain a duality for all Banach order-unit spaces or all Banach base-norm spaces, but not both at the same time., Comment: In Proceedings QPL 2018, arXiv:1901.09476

Resumen: En el presente artículo se analiza el desarrollo de un caso de defensa penal realizado en La Araucanía de Chile, en el que se condenó a un hombre mapuche por femicidio, el cual contó con la defensa especializada de la Defensoría Penal Mapuche el año 2013. A través del caso nos preguntamos en qué consistió la defensa realizada, de qué modo encaró la singularidad sociocultural, qué marcos de referencia teórico-metodológicos y recursos jurídicos incorporó en su argumentación y qué límites internos tuvo dicha práctica en el orden del convencimiento jurídico. El análisis considera las acciones de la defensa, la argumentación y la asimilación de categorías jurídicas exculpatorias y categorías de comprensión provenientes del universo cognitivo y médico mapuche. El valor de este caso radica en que, a través de él, es posible reconocer una forma de defensa penal sociocultural y visualizar los límites y potenciales pedagógicos que ésta tiene para una teoría y práctica penal pluralista y transformadora en contextos multiétnicos y multiculturales como el chileno actual. Abstract: This article analyzes the development of a criminal defense case that took place in Chile’s Araucanía Region in 2013, and in which a Mapuche man was convicted of femicide. The man’s legal defense was provided by the specialized Mapuche Public Defender’s office. In this article we examine the legal defense provided to this man, the manner in which the specific socio-cultural conditions were taken into account, the theoretical and methodological frameworks and juridical resources considered in the defense’s arguments, and the internal limitations met by said arguments in terms of the judicial convic- tion. Our analysis covers the defense’s actions, the arguments presented by the defense, and the assimilation of exculpatory legal categories and of categories of understanding that are typical of the Mapuche cognitive and medical worldview. This case is especially interesting because it allows us to recognize a specific type of socio-cultural legal defense and to derive lessons that can inform a pluralistic and fair legal theory and practice in multi-ethnic and multi-cultural contexts such as the Chilean society today.

This paper describes an equivalence of the canonical category of N -manifolds of degree 2 with a category of involutive double vector bundles. More precisely, we show how involutive double vector bundles are in duality with double vector bundles endowed with a linear metric. We describe then how special sections of the metric double vector bundle that is dual to a given involutive double vector bundle are the generators of a graded manifold of degree 2 over the double base. We discuss how split Poisson N -manifolds of degree 2 are equivalent to self-dual representations up to homotopy and so, following Gracia-Saz and Mehta, to linear splittings of a certain class of VB-algebroids. In other words, the equivalence of categories above induces an equivalence between so called Poisson involutive double vector bundles, which are the dual objects to metric VB-algebroids, and Poisson N -manifolds of degree 2.

For any ring R, the Auslander–Gruson–Jensen functor D A : fp ( R - Mod , Ab ) → ( mod - R , Ab ) op is the exact functor which sends a representable functor ( X , − ) to the tensor functor − ⊗ X . We show that this functor admits a fully faithful right adjoint D R and a fully faithful left adjoint D L . That is, we show that D A is part of a recollement of abelian categories. In particular, this shows that D A is a localisation and a colocalisation which gives an equivalence of categories fp ( R - Mod , Ab ) { F : D A F = 0 } ≃ ( mod - R , Ab ) op . We show that { F : D A F = 0 } is the Serre subcategory of fp ( R - Mod , Ab ) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R - Mod → Ab is also finitely presented.

The aim of this note is to analyse the structure of the $$L^0$$ -normed $$L^0$$ -modules over a metric measure space. These are a tool that has been introduced by Gigli to develop a differential calculus on spaces verifying the Riemannian Curvature Dimension condition. More precisely, we discuss under which conditions an $$L^0$$ -normed $$L^0$$ -module can be viewed as the space of sections of a suitable measurable Banach bundle and in which sense such correspondence can be actually made into an equivalence of categories.

In this paper we continue the investigations of the relationship between T 0 -quasi-metric spaces and partially ordered metric spaces. Among other things, we establish an equivalence of categories between so-called maximal T 0 -quasi-metric spaces and partially ordered metric spaces produced by T 0 -quasi-metrics. In the linear context we give geometric interpretations of the obtained results. In particular, we also derive a representation theorem for injective asymmetrically normed spaces.

We construct a monoidal category C w , v which categorifies the doubly-invariant algebra C N ′ ( w ) [ N ] N ( v ) associated with Weyl group elements w and v. It gives, after a localization, the coordinate algebra C [ R w , v ] of the open Richardson variety associated with w and v. The category C w , v is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v = id , C w , v is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra A q ( n ( w ) ) Z [ q , q − 1 ] given by Kang–Kashiwara–Kim–Oh. We show that the category C w , v contains special determinantial modules M ( w ≤ k Λ , v ≤ k Λ ) for k = 1 , … , l ( w ) , which commute with each other. When the quiver Hecke algebra R is symmetric, we find a formula of the degree of R-matrices between the determinantial modules M ( w ≤ k Λ , v ≤ k Λ ) . When it is of finite ADE type, we further prove that there is an equivalence of categories between C w , v and C u for w , u , v ∈ W with w = v u and l ( w ) = l ( v ) + l ( u ) .

We explore a conjecture of Morel about the Bass-Tate transfers defined on the contraction of a homotopy sheaf and prove that the conjecture is true with rational coefficients. Moreover, we study the relations between (contracted) homotopy sheaves, sheaves with Morel generalized transfers and Milnor-Witt homotopy sheaves, and prove an equivalence of categories. As applications, we describe the essential image of the canonical functor that forgets Milnor-Witt transfers and use these results to discuss the conservativity conjecture in motivic homotopy theory due to Bachmann and Yakerson.

Circuit algebras, introduced by Bar-Natan and the first author, are a generalization of Jones's planar algebras, in which one drops the planarity condition on "connection diagrams". They provide a useful language for the study of virtual and welded tangles in low-dimensional topology. In this note, we present the circuit algebra analogue of the well-known classification of planar algebras as pivotal categories with a self-dual generator. Our main theorem is that there is an equivalence of categories between circuit algebras and the category of linear wheeled props - a type of strict symmetric tensor category with duals that arises in homotopy theory, deformation theory and the Batalin-Vilkovisky quantization formalism., Comment: 29 pages, many figures

The $\otimes^\star$-monoidal structure on the category of sheaves on the $\mathrm{Ran}$ space is not pro-nilpotent in the sense of Francis-Gaitsgory. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the $\mathrm{Ran}$ space and integrating along the $\mathrm{Ran}$ space, i.e. taking factorization homology. Based on ideas sketched by Gaitsgory, we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah-Bott formula given in Gaitsgory-Lurie and Gaitsgory.

Let ${\mathbb{k}}$ be an algebraically closed field of characteristic p > 0 and let G be a connected reductive algebraic group over ${\mathbb{k}}$. Under some standard hypothesis on G, we give a direct approach to the finite W-algebra $U(\mathfrak{g},e)$ associated to a nilpotent element $e \in \mathfrak{g} = \textrm{Lie}\ G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the p-centre of $U(\mathfrak{g},e)$, which allows us to define reduced finite W-algebras $U_{\eta}(\mathfrak{g},e)$ and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabin’s equivalence of categories, generalizing recent work of the second author.

In this paper, we establish some connections between the concept of an equivalence of categories and that of an equivalence in a bicategory. Its main result builds upon the observation that two closely related concepts, which could both play the role of an equivalence in a bicategory, turn out not to coincide. Two counterexamples are provided for that goal, and detailed proofs are given. In particular, all calculations done in a bicategory are fully explicit, in order to overcome the difficulties which arise when working with bicategories instead of 2-categories.

The paper contains a survey of the current state of a constructive part in dessin d’enfants theory. Namely, it is devoted to actual establishing the correspondence between the Belyi pairs and their combinatorial-topological representations. This correspondence is established in terms of equivalence of categories, and all relevant categories are introduced. Several connections with arithmetic are discussed. One of the sections presents a possible generalization of the theory, in which three branch points, allowed for the Belyi functions, are replaced by four. Several directions for further research are presented. Bibliography: 80 titles.

If quantum mechanics were to be applicable to macroscopic objects, classical mechanics would have to be a limiting case of quantum mechanics. Then the category Set that packages classical mechanics has to be in some sense a 'limiting case' of the category Hilb packaging quantum mechanics. Following from this assumption, quantum-classical correspondence can be considered as a mapping of the category Hilb to the category Set, i.e., a functor from Hilb to Set, taking place in the macroscopic limit. As a procedure, which takes us from an object of the category Hilb (i.e., a Hilbert space) in the macroscopic limit to an object of the category Set (i.e., a set of values that describe the configuration of a system), this functor must take a finite number of steps in order to make the equivalence of Hilb and Set verifiable. However, as it is shown in the present paper, such a constructivist requirement cannot be met in at least one case of an Ising model of a spin glass. This could mean that it is impossible to demonstrate the emergence of classicality totally from the formalism of standard quantum mechanics., Major revision of the previous draft; 8 pages

This note does not claim anything new, since the material exposed here is somehow folkloric. We provide the main steps in showing the equivalence of categories between the category of BOCSES over a small linear category and the category of corings over the associated ring with enough orthogonal idempotents.