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Localization Of An ∞-category
In mathematics, specifically in higher category theory, a localization of an ∞-category is an ∞-category obtained by inverting some maps. An ∞-category is a presentable ∞-category if it is a localization of an ∞-presheaf category in the sense of Bousfield, by definition or as a result of Simpson. Definition Let ''S'' be a simplicial set and ''W'' a simplicial subset of it. Then the localization in the sense of Dwyer–Kan is a map :u: S \to W^S such that * W^S is an ∞-category, * the image u(W_1) consists of invertible maps, * the induced map on ∞-categories *:u^* : \operatorname(W^S, -) \overset\to \operatorname_W(S, -) :is invertible. When ''W'' is clear form the context, the localized category S^ W is often also denoted as L(S). A Dwyer–Kan localization that admits a right adjoint is called a localization in the sense of Bousfield. For example, the inclusion ∞-Grpd \hookrightarrow ∞-Cat has a left adjoint given by the localization that inverts all maps (f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Category Theory
In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic Invariant (mathematics), invariants of topological space, spaces, such as the Fundamental groupoid, fundamental . In higher category theory, the concept of higher categorical structures, such as (), allows for a more robust treatment of homotopy theory, enabling one to capture finer homotopical distinctions, such as differentiating two topological spaces that have the same fundamental group but differ in their higher homotopy groups. This approach is particularly valuable when dealing with spaces with intricate topological features, such as the Eilenberg-MacLane space. Strict higher categories An ordinary category (m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
∞-category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a Category (mathematics), category. The study of such generalizations is known as higher category theory. Overview Quasi-categories were introduced by . André Joyal has much advanced the study of quasi-categories showing that most of the usual basic category theory and some of the advanced notions and theorems have their analogues for quasi-categories. An elaborate treatise of the theory of quasi-categories has been expounded by . Quasi-categories are certain simplicial sets. Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). But unlike categories, the composition of two morphisms need not be uniquely defined. All the morphisms that can serve as composition of tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Core Of A Category
In mathematics, especially category theory, the core of a category ''C'' is the category whose objects are the objects of ''C'' and whose morphisms are the invertible morphisms in ''C''.Pierre Gabriel, Michel Zisman, § 1.5.4., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967/ref> In other words, it is the largest groupoid subcategory. As a functor C \mapsto \operatorname(C), the core is a right adjoint to the inclusion of the category of (small) groupoids into the category of (small) categories. On the other hand, the left adjoint to the above inclusion is the fundamental groupoid functor. For ∞-categories, \operatorname is defined as a right adjoint to the inclusion ∞-Grpd \hookrightarrow ∞-Cat. The core of an ∞-category C is then the largest ∞-groupoid contained in C. The core of ''C'' is also often written as C^. The left adjoint to the above inclusion is given by a localization of an ∞-category. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hereditary ∞-category
Heredity, also called inheritance or biological inheritance, is the passing on of traits from parents to their offspring; either through asexual reproduction or sexual reproduction, the offspring cells or organisms acquire the genetic information of their parents. Through heredity, variations between individuals can accumulate and cause species to evolve by natural selection. The study of heredity in biology is genetics. Overview In humans, eye color is an example of an inherited characteristic: an individual might inherit the "brown-eye trait" from one of the parents. Inherited traits are controlled by genes and the complete set of genes within an organism's genome is called its genotype. The complete set of observable traits of the structure and behavior of an organism is called its phenotype. These traits arise from the interaction of the organism's genotype with the environment. As a result, many aspects of an organism's phenotype are not inherited. For example, sun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
∞-topos
In mathematics, an ∞-topos (infinity-topos) is, roughly, an ∞-category such that its objects behave like sheaf (mathematics), sheaves of spaces with some choice of Grothendieck topology; in other words, it gives an intrinsic notion of sheaves without reference to an external space. The prototypical example of an ∞-topos is the ∞-category of sheaves of spaces on some topological space. But the notion is more flexible; for example, the ∞-category of étale sheaves on some Scheme (mathematics), scheme is not the ∞-category of sheaves on any topological space but it is still an ∞-topos. Precisely, in Lurie's ''Higher Topos Theory'', an ∞-topos is defined as an ∞-category ''X'' such that there is a small ∞-category ''C'' and an (accessible ∞-category, accessible) left exact localization of an ∞-category, localization functor from the ∞-category of presheaf of spaces, presheaves of spaces on ''C'' to ''X''. A theorem of Lurie states that an ∞-category is an � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
