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Model Categories
In mathematics, particularly in homotopy theory, a model category is a category theory, category with distinguished classes of morphisms ('arrows') called 'weak equivalence (homotopy theory), weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic K-theory, algebraic ''K''-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results. Motivation Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
