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Left Proper Model Structure
In higher category theory in mathematics, a proper model structure is a model structure in which additionally weak equivalences are preserved under pullback (fiber product) along fibrations, called ''right proper'', and pushouts (cofiber product) along cofibrations, called ''left proper''. It is helpful to construct weak equivalences and hence to find isomorphic objects in the homotopy theory of the model structure. Definition For every model category, one has: * Pushouts of weak equivalences between cofibrant objects along cofibrations are again weak equivalences. * Pullbacks of weak equivalences between fibrant objects along fibrations are again weak equivalences. A model category is then called: * ''left proper'', if pushouts of weak equivalences along cofibrations are again weak equivalences. * ''right proper'', if pullbacks of weak equivalences along fibrations are again weak equivalences. * ''proper'', if it is both left proper and right proper. Properties * A mode ... [...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] |
Pullback (category Theory)
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit (category theory), limit of a diagram (category theory), diagram consisting of two morphisms and with a common codomain. The pullback is written :. Usually the morphisms and are omitted from the notation, and then the pullback is written :. The pullback comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative diagram, commutative square. The Dual (category theory), dual concept of the pullback is the ''Pushout ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pushout (category Theory)
In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms ''f'' : ''Z'' → ''X'' and ''g'' : ''Z'' → ''Y'' with a common domain. The pushout consists of an object ''P'' along with two morphisms ''X'' → ''P'' and ''Y'' → ''P'' that complete a commutative square with the two given morphisms ''f'' and ''g''. In fact, the defining universal property of the pushout (given below) essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are P = X \sqcup_Z Y and P = X +_Z Y. The pushout is the categorical dual of the pullback. Universal property Explicitly, the pushout of the morphisms ''f'' and ''g'' consists of an object ''P'' and two morphisms ''i''1 : ''X'' → ''P'' and ''i''2 : ''Y'' → ''P'' such that the diagram : commutes and such th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline. Applications to other fields of mathematics Besides algebraic topology, the theory has also been used in other areas of mathematics such as: * Algebraic geometry (e.g., A1 homotopy theory, A1 homotopy theory) * Category theory (specifically the study of higher category theory, higher categories) Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid Pathological (mathematics), pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being Category of compactly generated weak Hausdorff spaces, compactly generated weak Hausdorff or a CW complex. In the same vein as above, a "Map (mathematics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Category
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), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quillen Adjunction
In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen. Formal definition Given two closed model categories C and D, a Quillen adjunction is a pair :(''F'', ''G''): C \leftrightarrows D of adjoint functors with ''F'' left adjoint to ''G'' such that ''F'' preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that ''G'' preserves fibrations and trivial fibrations. In such an adjunction ''F'' is called the left Quillen functor and ''G'' is called the right Quillen functor. Properties It is a consequence of the axioms that a left (right) Quillen functor preserves weak equivalences between cofibrant (fibrant) objects. The total derived funct ... [...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] |
Mathematical Surveys And Monographs
''Mathematical Surveys and Monographs'' is a series of monographs published by the American Mathematical Society. Each volume in the series gives a survey of the subject along with a brief introduction to recent developments and unsolved problems. The series has been known as ''Mathematical Surveys and Monographs'' since 1984. Its ISSN An International Standard Serial Number (ISSN) is an eight-digit to uniquely identify a periodical publication (periodical), such as a magazine. The ISSN is especially helpful in distinguishing between serials with the same title. ISSNs a ... is 0885–4653. The series was founded in 1943 as 'Mathematical Surveys'. External links External links Mathematical Surveys and Monographs Publications of the American Mathematical Society Publications established in 1943 Monographic series Series of mathematics books {{mathematics-lit-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
