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β-topos
In mathematics, an β-topos (infinity-topos) is, roughly, an β-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 β-topos is the β-category of sheaves of spaces on some topological space. But the notion is more flexible; for example, the β-category of Γ©tale sheaves on some Scheme (mathematics), scheme is not the β-category of sheaves on any topological space but it is still an β-topos. Precisely, in Lurie's ''Higher Topos Theory'', an β-topos is defined as an β-category ''X'' such that there is a small β-category ''C'' and an (accessible β-category, accessible) left exact localization of an β-category, localization functor from the β-category of presheaf of spaces, presheaves of spaces on ''C'' to ''X''. A theorem of Lurie states that an β-category is an οΏ½ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion of localization. The Grothendieck topoi find applications in algebraic geometry, and more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory. Grothendieck topos (topos in geometry) Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this progra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of An β-category
In mathematics, specifically in higher category theory, a localization of an β-category is an β-category obtained by inverting some maps. An β-category is a presentable β-category if it is a localization of an β-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 DwyerβKan is a map :u: S \to W^S such that * W^S is an β-category, * the image u(W_1) consists of invertible maps, * the induced map on β-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 DwyerβKan localization that admits a right adjoint is called a localization in the sense of Bousfield. For example, the inclusion β-Grpd \hookrightarrow β-Cat has a left adjoint given by the localization that inverts all maps (f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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] |
Foundations Of Mathematics
Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theorems, proof (mathematics), proofs, algorithms, etc. in particular. This may also include the philosophy of mathematics, philosophical study of the relation of this framework with reality. The term "foundations of mathematics" was not coined before the end of the 19th century, although foundations were first established by the ancient Greek philosophers under the name of Aristotle's logic and systematically applied in Euclid's Elements, Euclid's ''Elements''. A mathematical assertion is considered as truth (mathematics), truth only if it is a theorem that is proved from true premises by means of a sequence of syllogisms (inference rules), the premises being either already proved theorems or self-evident assertions called axioms or postulat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Set
In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Simplic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bousfield Localization
In category theory, a branch of mathematics, a (left) Bousfield localization of a model category replaces the model structure with another model structure with the same cofibrations but with more weak equivalences. Bousfield localization is named after Aldridge Bousfield, who first introduced this technique in the context of localization of topological spaces and spectra. Model category structure of the Bousfield localization Given a class ''C'' of morphisms in a model category ''M'' the left Bousfield localization is a new model structure on the same category as before. Its equivalences, cofibrations and fibrations, respectively, are * the ''C''-local equivalences * the original cofibrations of ''M'' and (necessarily, since cofibrations and weak equivalences determine the fibrations) * the maps having the right lifting property with respect to the cofibrations in ''M'' which are also ''C''-local equivalences. In this definition, a ''C''-local equivalence is a map f\colon X \to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Effective Groupoid
Effectiveness or effectivity is the capability of producing a desired result or the ability to produce desired output. When something is deemed effective, it means it has an intended or expected outcome, or produces a deep, vivid impression. Etymology The origin of the word ''effective'' stems from the Latin word , which means "creative, productive, or effective". It surfaced in Middle English between 1300 and 1400 AD. Usage Science and technology Mathematics and logic In mathematics and logic, ''effective'' is used to describe metalogical methods that fit the criteria of an effective procedure. In group theory, a group element acts ''effectively'' (or ''faithfully'') on a point, if that point is not fixed by the action. Physics In physics, an effective theory is, similar to a phenomenological theory, a framework intended to explain certain (observed) effects without the claim that the theory correctly models the underlying (unobserved) processes. In heat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presentable β-category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, β-category, Boardman complex, quategory) is a generalization of the notion of a category. The study of such generalizations is known as higher category theory. Overview Quasi-categories were introduced by . AndrΓ© 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 two given morphisms are r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Topos
Ordinary or The Ordinary often refer to: Music * ''Ordinary'' (EP) (2015), by South Korean group Beast * ''Ordinary'' (album) (2011), by Every Little Thing * "Ordinary" (Alex Warren song) (2025) * "Ordinary" (Two Door Cinema Club song) (2016) * "Ordinary" (Wayne Brady song) (2008) * "Ordinary", song by Train from '' Alive at Last'' (2004) Religion * Ordinary (Catholic Church), a supervisor, typically a bishop, in charge of a territory comparable to a diocese, or a major superior of a religious institute * Ordinary (church officer), an officer of a church or civic authority who by reason of office has ordinary power to execute laws * Ordinary (liturgy), a set of texts in Roman Catholic and other Western Christian liturgies that are generally invariable * Ordinary Time, the parts of the Roman Catholic liturgical year that are outside Advent, Christmastide, Lent, and Eastertide. * Ordinary (lecture), a type of lecture given in universities of the Middle Ages Other * An arch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Accessible β-category
In mathematics, especially category theory, an accessible quasi-category is a quasi-category in which each object is an ind-object on some small quasi-category. In particular, an accessible quasi-category is typically large (not small). The notion is a generalization of an earlier 1-category version of it, an accessible category introduced by AdΓ‘mek and RosickΓ½. Definition An β-category In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, β-category, Boardman complex, quategory) is a generalization of the notion of a Category (ma ... is called accessible or more precisely \kappa-accessible if it is equivalent to the β-category of \kappa-ind objects on some small β-category for some regular cardinal \kappa. Facts A small β-category is accessible if and only if it is idempotent-complete. References * * Charles Rezk, Generalizing accessible β-categories, 202dr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
