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Cisinski Model Structure
In higher category theory in mathematics, a Cisinski model structure is a special kind of model structure on topoi. In homotopical algebra, the category of simplicial sets is of particular interest. Cisinski model structures are named after Denis-Charles Cisinski, who introduced them in 2001. His work is based on unfinished ideas presented by Alexander Grothendieck in his script ''Pursuing Stacks'' from 1983. Definition A cofibrantly generated model structure on a topos, for the cofibrations are exactly the monomorphisms, is called a ''Cisinski model structure''. Cofibrantly generated means that there are small sets I and J of morphisms, on which the small object argument can be applied, so that they generate all cofibrations and trivial cofibrations using the lifting property: : \operatorname =^\perp(I^\perp); : W\cap\operatorname =^\perp(J^\perp); More generally, a small set generating the class of monomorphisms of a category of presheaves is called ''cellular model'': : \ ... [...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] |
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] |
Model Structure
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided into physical models (e.g. a ship model or a fashion model) and abstract models (e.g. a set of mathematical equations describing the workings of the atmosphere for the purpose of weather forecasting). Abstract or conceptual models are central to philosophy of science. In scholarly research and applied science, a model should not be confused with a theory: while a model seeks only to represent reality with the purpose of better understanding or predicting the world, a theory is more ambitious in that it claims to be an explanation of reality. Types of model ''Model'' in specific contexts As a noun, ''model'' has specific meanings in certain fields, derived from its original meaning of "structural design or layout": * Model (art), a person ... [...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] |
Category Of Simplicial Sets
Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *Category (Vaisheshika) * Stoic categories *Category mistake Science *Cognitive categorization, categories in cognitive science *Statistical classification, statistical methods used to effect classification/categorization Mathematics * Category (mathematics), a structure consisting of objects and arrows * Category (topology), in the context of Baire spaces * Lusternik–Schnirelmann category, sometimes called ''LS-category'' or simply ''category'' * Categorical data, in statistics Linguistics *Lexical category, a part of speech such as ''noun'', ''preposition'', etc. *Syntactic category, a similar concept which can also include phrasal categories *Grammatical category, a grammatical feature such as ''tense'', ''gender'', etc. Other * Category (chess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Denis-Charles Cisinski
Denis-Charles Cisinski (born March 10, 1976) is a mathematician focussing on higher category theory, homotopy theory, K-theory and algebraic geometry. In 2001, Cisinski model structures on topoi were introduced and later named after him. Since 2016, Denis-Charles Cisinski works at the Universität Regensburg. Research Denis-Charles Cisinski obtained his PhD in 2002 at the Paris Diderot University with a thesis supervised by Georges Maltsiniotis and titled ''Les préfaisceaux comme modèles des types d'homotopie'' ( Presheaves as models for homotopy types). It was expanded and released as a book in 2006, further developing the theory from ''Pursuing Stacks'' by Alexander Grothendieck. In 2015, Denis-Charles Cisinski gave a talk at the Séminaire Nicolas Bourbaki summarizing the current state of research titled ''Catégories supérieures et théorie des topos'' (Higher categories and theory of toposes). '' Higher Categories and Homotopical Algebra'', a mathematical textbook about ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pursuing Stacks
''Pursuing Stacks'' () is an influential 1983 mathematical manuscript by Alexander Grothendieck. It consists of a 12-page letter to Daniel Quillen followed by about 600 pages of research notes. The topic of the work is a generalized homotopy theory using higher category theory. The word "stacks" in the title refers to what are nowadays usually called " ∞-groupoids", one possible definition of which Grothendieck sketches in his manuscript. (The stacks of algebraic geometry, which also go back to Grothendieck, are not the focus of this manuscript.) Among the concepts introduced in the work are derivators and test categories. Some parts of the manuscript were later developed in: * * Overview of manuscript I. The letter to Daniel Quillen Pursuing stacks started out as a letter from Grothendieck to Daniel Quillen. In this letter he discusses Quillen's progress on the foundations for homotopy theory and remarked on the lack of progress since then. He remarks how some of his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Object Argument
In mathematics, especially in category theory, Quillen’s small object argument, when applicable, constructs a factorization of a morphism in a functorial way. In practice, it can be used to show some class of morphisms constitutes a weak factorization system in the theory of model categories. The argument was introduced by Quillen to construct a model structure on the category of (reasonable) topological spaces. The original argument was later refined by Garner. Statement Let C be a category that has all small colimits. We say an object x in it is compact with respect to an ordinal \omega if \operatorname(x, -) commutes with an \omega-filterted colimit. In practice, we fix \omega and simply say an object is compact if it is so with respect to that fixed \omega. If F is a class of morphismms, we write l(F) for the class of morphisms that satisfy the left lifting property with respect to F. Similarly, we write r(F) for the right lifting property. Then Example: presheaf Her ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lifting Property
In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given class of morphisms. It appears in a prominent way in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen. It is also used in the definition of a factorization system, and of a weak factorization system, notions related to but less restrictive than the notion of a model category. Several elementary notions may also be expressed using the lifting property starting from a list of (counter)examples. Formal definition A morphism i in a category has the ''left lifting property'' with respect to a morphism p, and p also has the ''right lifting property'' with respect to i, sometimes denoted i\perp p or i\downarrow p, iff the following implication holds for each morphism f and g in the cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presheaves
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well-behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every datum is the sum of its constituent data). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their precise definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joyal Model Structure
In higher category theory in mathematics, the Joyal model structure is a special model structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called ''fibrations'', ''cofibrations'' and ''weak equivalences'', which fulfill the properties of a model structure. Its fibrant objects are all ∞-categories and it furthermore models the homotopy theory of CW complexes up to homotopy equivalence, with the correspondence between simplicial sets and CW complexes being given by the geometric realization and the singular functor. The Joyal model structure is named after André Joyal. Definition The Joyal model structure is given by: * Fibrations are isofibrations.Cisinski 2019, Theorem 3.6.1. * Cofibrations are monomorphisms.Lurie 2009, ''Higher Topos Theory'', Theorem 1.3.4.1. * Weak equivalences are ''weak categorical equivalences'',Joyal 2008, Theorem 6.12. hence morphisms between simplicial sets, whose geometric realization is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kan–Quillen Model Structure
In higher category theory, the Kan–Quillen model structure is a special model structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called ''fibrations'', ''cofibrations'' and ''weak equivalences'', which fulfill the properties of a model structure. Its fibrant objects are all Kan complexes and it furthermore models the homotopy theory of CW complexes up to weak homotopy equivalence, with the correspondence between simplicial sets, Kan complexes and CW complexes being given by the geometric realization and the singular functor ( Milnor's theorem). The Kan–Quillen model structure is named after Daniel Kan and Daniel Quillen. Definition The Kan–Quillen model structure is given by: * Fibrations are Kan fibrations.Joyal 2008, Theorem 6.1. on p. 293 * Cofibrations are monomorphisms.Cisinski 2019, Theorem 3.1.8. * Weak equivalences are ''weak homotopy equivalences'', hence morphisms between simplicial sets, whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
