![]() The second construction gives an indication that one can possibly develop a noncommutative proper homotopy theory in the context of topological algebras, e.g., pro C ∗-algebras. The first result can be used to deduce derived Morita equivalence between DG categories of topological bundles associated to separable C ∗-algebras up to a K-theoretic identification from the knowledge of KK-equivalence between the C ∗-algebras. Whats an example of a locally presentable category in nature thats not aleph0-locally presentable The category of Banach spaces and contractions. This construction respects homotopy between proper maps after enforcing matrix stability on the category of pro C ∗-algebras. Generation from generators The notion of locally presentable category is, at least roughly, an analogue for categories of the notion of a finitely generated module. The properties of these categories are studied extensively, in particular their close relationship with other types of categories. Motivated by a construction of Cuntz we associate a pro C ∗-algebra to any simplicial set, which is functorial with respect to proper maps of simplicial sets and those of pro C ∗-algebras. The general idea is that a locally presentable category is a large category generated from small data: from small objects under small colimit. The notions of a locally -presentable and locally -generated categories are introduced, where is a regular cardinal. Locally presentable and accessible categories.We construct an additive functor from the category of separable C ∗-algebras with morphisms enriched over Kasparov’s KK0-groups to the noncommutative correspondence category NCC K dg, whose objects are small DG categories and morphisms are given by the equivalence classes of some DG bimodules up to a certain K-theoretic identification. For researchers in category theory, algebra, computer science, and model theory, this book will be a necessary purchase. In the final chapters they treat some topics in model theory and some set theoretical aspects. The authors go on to study categories of algebras and prove that Freyd's essentially algebraic categories are precisely the locally presentable categories. Firstly the properties of l-presentable objects, locally l-presentable categories, and l-accessible categories are discussed in detail, and the equivalence of accessible and sketchable categories is proved. It was further generalized by Makkai and Paré who introduced accessible categories in the monograph 20 which convincingly demonstrated the importance of this notion. The aim of this book is to provide an exposition of both the theory and the applications of these categories at a level accessible to graduate students. Introduction The concept of a locally presentable category was introduced by Gabriel and Ulmer 18. The concepts of a locally presentable category and an accessible category have turned out to be useful in formulating connections between universal algebra, model theory, logic and computer science. Includes bibliographical references (pages 299-307) and index. Catégories (Mathématiques) Kategorientheorie Electronic books. Sheaves with values in a locally presentable closed symmetric monoidal category. Categorieën (wiskunde) Lógica matemática. The category of convenient vector spaces is locally presentable (see 38) and closed symmetric monoidal 26. Categories (Mathematics) Representations of categories. London Mathematical Society lecture note series 189 London Mathematical Society lecture note series 189.Ĭategories (Mathematics) Representations of categories. Saved in: Bibliographic Details Author / Creator:Ĭambridge New York, NY : Cambridge University Press, 1994.
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