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Simplicial Localization
In category theory, a branch of mathematics, the simplicial localization of a category ''C'' with respect to a class ''W'' of morphisms of ''C'' is a simplicial category ''LC'' whose \pi_0 is the localization C ^/math> of ''C'' with respect to ''W''; that is, \pi_0 LC(x, y) = C ^x, y) for any objects ''x'', ''y'' in ''C''. The notion is due to Dwyer and Kan. References *W. G. Dwyer and Dan KanSimplicial localizations of categories *http://math.mit.edu/~mdono/_Juvitop.pdf External links * Category theory {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. '' Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicially Enriched Category
In mathematics, a simplicially enriched category, is a category (mathematics), category enriched category, enriched over the category of simplicial sets. Simplicially enriched categories are often also called, more ambiguously, simplicial categories; the latter term however also applies to simplicial objects in Cat (the category of small categories). Simplicially enriched categories can, however, be identified with simplicial objects in Cat whose object part is constant, or more precisely, whose all face and degeneracy maps are bijective on objects. Simplicially enriched categories can model (∞, 1)-category, (∞, 1)-categories, but the dictionary has to be carefully built. Namely many notions, limits for example, are different from the limits in the sense of enriched category theory. References * * External links * {{Category theory Category theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Category
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category. Introduction and motivation A category ''C'' consists of objects and morphisms between these objects. The morphisms reflect relations between the objects. In many situations, it is meaningful to replace ''C'' by another category ''C in which certain morphisms are forced to be isomorphisms. This process is called localization. For example, in the category of ''R''-modules (for some fixed commutative ring ''R'') the multiplication by a fixed element ''r'' of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dan Kan
Daniel Marinus Kan (or simply Dan Kan) (August 4, 1927 – August 4, 2013) was a Dutch mathematician working in category theory and homotopy theory. He was a prolific contributor to both fields for six decades, having authored or coauthored several dozen research papers and monographs. Career He received his Ph.D. at Hebrew University in 1955, under the direction of Samuel Eilenberg. His students include Aldridge K. Bousfield, William Dwyer, Stewart Priddy, Emmanuel Dror Farjoun and Jeffrey H. Smith. He was an emeritus professor at the Massachusetts Institute of Technology where he taught from 1959, formally retiring in 1993. Work He played a role in the beginnings of modern homotopy theory similar to that of Saunders Mac Lane in homological algebra, namely the adroit and persistent application of categorical methods. His most famous work is the abstract formulation of the discovery of adjoint functors, which dates from 1958. The Kan extension is one of the broadest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
