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Fibration Of Simplicial Sets
In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions \Lambda^n_i \subset \Delta^n, 0 \le i < n. A right fibration is defined similarly with the condition . A is one with the right lifting property with respect to every horn inclusion; hence, a Kan fibration is exactly a map that is both a left and right fibration. Examples A right fibration is a such that each fiber is a . In particular, a[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Cofibration
In mathematics, in particular homotopy theory, a continuous mapping between topological spaces :i: A \to X, is a ''cofibration'' if it has the homotopy extension property with respect to all topological spaces S. That is, i is a cofibration if for each topological space S, and for any continuous maps f, f': A\to S and g:X\to S with g\circ i=f, for any homotopy h : A\times I\to S from f to f', there is a continuous map g':X \to S and a homotopy h': X\times I \to S from g to g' such that h'(i(a),t)=h(a,t) for all a\in A and t\in I. (Here, I denotes the unit interval ,1/math>.) This definition is formally dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces; this is one instance of the broader Eckmann–Hilton duality in topology. Cofibrations are a fundamental concept of homotopy theory. Quillen has proposed the notion of model category as a formal framework for doing homotopy theory in more general categories; a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, '' The Daily Princetonian'', and later added book publishing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Universal Cartesian Fibration
Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company that is a subsidiary of Comcast ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a television channel owned by NBCUniversal ** Universal Kids, an American current television channel, formerly known as Sprout, owned by NBCUniversal ** Universal Pictures, an American film studio, and a subsidiary of NBCUniversal ** Universal Television, a television division owned by NBCUniversal Content Studios ** Universal Destinations & Experiences, the theme park unit of NBCUniversal * Universal Airlines (other) * Universal Avionics, a manufacturer of flight control components * Universal Corporation, an American tobacco company * Universal Display Corporation, a manufacturer of displays * Universal Edition, a classical music publishing firm, founded in Vienna in 1901 * Universal Entertainme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
