On Representations of Grothendieck Toposes

1987 ◽  
Vol 39 (1) ◽  
pp. 168-221 ◽  
Author(s):  
Michael Barr ◽  
Michael Makkai
Keyword(s):  
Representation Theory ◽  
Left Adjoint ◽  
Typical Result ◽  
Topos Theory ◽  
Exact Functor ◽  
Continuous Embedding ◽  
The Subject ◽  
Geometric Morphism ◽  
Exact Categories

Results of a representation-theoretic nature have played a major role in topos theory since the beginnings of the subject. For example, Deligne's theorem on coherent toposes, which says that every coherent topos has a continuous embedding into a topos of the form SetI for a discrete set I, is a typical result in the representation theory of toposes. (A continuous functor between toposes is the left adjoint of a geometric morphism. For Grothendieck toposes, it is exactly the same as a continuous functor between them, considered as sites with their canonical topologies. By a continuous functor between sites on left exact categories, we mean a left exact functor taking covers to covers.)A representation-like result for toposes typically asserts that a topos that satisfies some abstract conditions is related to a topos of some concrete kind; the relation between them is usually an embedding of the first topos in the second (concrete) one, for which the embedding satisfies some additional properties (fullness, etc.).

Simply Connected Limits

1990 ◽  
Vol 42 (4) ◽  
pp. 731-746 ◽  
Author(s):  
Robert Paré
Keyword(s):  
Left Adjoint ◽  
Topos Theory ◽  
Essential Ingredient ◽  
Original Definition ◽  
The Subject ◽  
Definition Of ◽  
Simply Connected ◽  
Geometric Morphisms ◽  
Exact Categories

The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

Model Theory: Geometrical and Set-Theoretic Aspects and Prospects

2003 ◽  
Vol 9 (2) ◽  
pp. 197-212 ◽  
Author(s):  
Angus Macintyre
Keyword(s):  
Model Theory ◽  
Category Theory ◽  
Theoretic Model ◽  
Topos Theory ◽  
Sheaf Theory ◽  
Special Cases ◽  
Algebraic Spaces ◽  
The Subject ◽  
Near Future ◽  
Modern Mathematics

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable (to the geometers), this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly.

Disjointly Additive Operators and Modular Spaces

1990 ◽  
Vol 42 (3) ◽  
pp. 508-519
Author(s):  
Iwo Labuda
Keyword(s):  
Additive Functional ◽  
Typical Result ◽  
Additive Functionals ◽  
Modular Spaces ◽  
The Subject ◽  
Image Position

By now the literature concerning the representation of disjointly additive functionals and operators is quite extensive. A few entries on the subject are [6, 7, 8, 11, 20, 21]. In [7, 8, 17] further references can be found, in [7] the “prehistory” of the subject is also discussed.To quote a typical result, we may take a 1967 theorem of Drewnowski and Orlicz ([6] Th. 3.2, [17] 12.4) which asserts that, under proper assumptions, an abstract modular (= disjointly countably additive functional) p on a “substantial“ subspace D of L° can be realized by the formula .

scholarly journals Complete slices and homological properties of tilted algebras

1994 ◽  
Vol 36 (3) ◽  
pp. 347-354 ◽  
Author(s):  
Ibrahim Assem ◽  
Flávio Ulhoa Coelho
Keyword(s):  
Representation Theory ◽  
Global Dimension ◽  
Indecomposable Modules ◽  
Tilted Algebra ◽  
Finite Dimensional ◽  
Tilted Algebras ◽  
Homological Dimensions ◽  
The Subject ◽  
Left And Right ◽  
Almost All

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

scholarly journals The Empirical Research of Everyday Anthropogenic Spatial-Object Environment Psychological Representation (in Case of Self-isolation)

2020 ◽  
pp. 64-70
Author(s):  
Yulia Gennadievna Panyukova
Keyword(s):  
Representation Theory ◽  
Environmental Stress ◽  
Everyday Life ◽  
Empirical Research ◽  
Environmental Psychology ◽  
Feature Analysis ◽  
Spatial Object ◽  
Stress Theory ◽  
The Subject ◽  
Drawing Technique

The study describes the results of empirical research, which is related to feature analysis of personal psychological representation of anhtropogenic spatial-objective environment of everyday life. Based on systematization of the theoretical provisions, developed in the psychological representation theory, in environmental psychology, in everyday life psychology and in the environmental stress theory, the psychologically relevant topological indicators of representation were identified. According to the data obtained as a result of the drawing technique “My life in self-isolation” and the description of the drawing, several types of representation were identified. Several indicators for drawing classification were determined: presence/absence of an image of respondent, presence/absence of the border between “internal” and “external” everyday environment as well as homogeneity/multidimensionality of this environment. These types demonstrate resource/deficiency of the environment of everyday life in conditions of self-isolation for the subject.

Representation theory for tensor products of Banach algebras

1981 ◽  
Vol 90 (3) ◽  
pp. 445-463 ◽  
Author(s):  
T. K. Carne
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Tensor Products ◽  
The Subject ◽  
Algebraic Tensor Product ◽  
Natural Way

The algebraic tensor product A1⊗A2 of two Banach algebras is an algebra in a natural way. There are certain norms α on this tensor product for which the multiplication is continuous so that the completion, A1αA2, is a Banach algebra. The representation theory of such tensor products is the subject of this paper. It will be shown that, under certain simple conditions, the tensor product of two semi-simple Banach algebras is semi-simple although, without these conditions, the result fails.

Anaphora

2019 ◽  
pp. 500-544
Author(s):  
Mary Dalrymple ◽  
John J. Lowe ◽  
Louise Mycock
Keyword(s):  
Representation Theory ◽  
Discourse Representation ◽  
Discourse Representation Theory ◽  
Different Types ◽  
Semantic Treatment ◽  
The Subject ◽  
Structural Relations ◽  
Anaphoric Relations

This chapter presents LFG analyses for different types of anaphora. Section 14.1 discusses how incorporated pronominal elements behave differently from elements that alternate with agreement markers, and the ways in which these differ from morphologically independent pronouns. Anaphoric relations and binding patterns have been the subject of much research within the LFG framework; Section 14.2 discusses positive and negative constraints on anaphoric binding stated in terms of structural relations holding at f-structure, and Section 14.3 discusses prominence relations which hold between the anaphor and its potential antecedents stated at f-structure as well as other linguistic levels. A glue-theoretic treatment of the semantics of anaphoric binding is presented in Section 14.4, modeled using a version of Discourse Representation Theory. This semantic treatment will be drawn upon in subsequent chapters, particularly in the discussion of anaphoric control in Chapter 15.

scholarly journals Aula em equipe como estratégia inovadora de ensino

2019 ◽  
Vol 82 (200-01-02) ◽  
Author(s):  
Alberto Merchede
Keyword(s):  
Representation Theory ◽  
Social Representation ◽  
Human Values ◽  
Research Theme ◽  
The Social ◽  
Challenging Research ◽  
Teaching Research ◽  
Social Representation Theory ◽  
The Subject

Retoma uma técnica alternativa de instrução centrada em grupo, trabalho já publicado, aperfeiçoando-o, revendo-o e ampliando-o, mediante introdução de modificações, fruto de avaliações realizadas no correr do tempo, desde a sua publicação na versão original. A técnica é calcada na experiência e corporificada num processo evolutivo de ajustes e correções. Inicia-se com a abordagem de alguns aspectos do seminário, em que a técnica encontrou sua idéia germinal. Destacam-se críticas sobre a freqüente má utilização do seminário. A seguir, expõe-se a Técnica de Aula em Equipe, sua definição, objetivos e suas diferentes etapas de aplicação: planejamento, preparação, apresentações e avaliação. São tecidas considerações a respeito da utilização da técnica, tais como a reação dos alunos; alguns aspectos incorporados à técnica como forma de correção de problemas anteriores ou impropriedades detectadas na utilização do seminário. Nas conclusões, apontam-se alguns resultados positivos do uso da técnica. Palavras-chave: dinâmica de grupo; método de ensino; seminário; técnica de ensino. Abstract The problem of "cheating on tests" is not only a polemic one, but is also full of controversy. This is so, when the subject is viewed both from a didactics and pupil evaluation perspective and from a human values perspective. Hence, "cheating on tests" constitutes a significant and challenging research theme. Thus, turning "cheating on tests" into a problem is to rethink it on a critical and strongly based way, which enables the perception and analysis of the several sides of its origin and circumstances. Bearing that on mind, that survey bases itself on the Social Representation Theory to discuss images, concepts, practices and tools as well as to study alternatives of that "problem", giving special attention to the interrelation between school and psychosocial reasons. Keywords: social representation; "cheating on tests"; teaching; research.

Quasicomponents in topos theory: the hyperpure, complete spread factorization

2007 ◽  
Vol 142 (1) ◽  
pp. 47-62
Author(s):  
MARTA BUNGE ◽  
JONATHON FUNK
Keyword(s):  
Connected Domain ◽  
Existence And Uniqueness ◽  
Topos Theory ◽  
Local Connectedness ◽  
Special Case ◽  
Geometric Morphism ◽  
Geometric Morphisms

AbstractWe establish the existence and uniqueness of a factorization for geometric morphisms that generalizes the pure, complete spread factorization for geometric morphisms with a locally connected domain. A complete spread with locally connected domain over a topos is a geometric counterpart of a Lawvere distribution on the topos, and the factorization itself is of the comprehensive type. The new factorization removes the topologically restrictive local connectedness requirement by working with quasicomponents in topos theory. In the special case when the codomain topos of the geometric morphism coincides with the base topos, the factorization gives the locale of quasicomponents of the domain topos, or its ‘0-dimensional’ reflection.

scholarly journals Finitary birepresentations of finitary bicategories

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Marco Mackaay ◽  
Volodymyr Mazorchuk ◽  
Vanessa Miemietz ◽  
Daniel Tubbenhauer ◽  
Xiaoting Zhang
Keyword(s):  
Representation Theory ◽  
Equivalence Classes ◽  
Double Centralizer ◽  
The Subject

Abstract In this paper, we discuss the generalization of finitary 2-representation theory of finitary 2-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive 2-representations of a given 2-category was reduced to that for certain subquotients. These reduction results were all formulated as bijections between equivalence classes of 2-representations. In this paper, we generalize them to biequivalences between certain 2-categories of birepresentations. Furthermore, we prove an analog of the double centralizer theorem in finitary birepresentation theory.

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