Functors | Homotopy theory | Simplicial sets | Algebraic topology

Simplicial set

In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a "well-behaved" topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces. Simplicial sets are used to define quasi-categories, a basic notion of higher category theory. A construction analogous to that of simplicial sets can be carried out in any category, not just in the category of sets, yielding the notion of simplicial objects. (Wikipedia).

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From playlist Mathematics

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From playlist Set Theory

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This video introduces the basic vocabulary used in set theory. http://mathispower4u.wordpress.com/

From playlist Sets

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From playlist Sets (Discrete Math)

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From playlist Set Theory

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http://mathispower4u.wordpress.com/

From playlist Integers

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From playlist Set Theory by Mathoma

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From playlist Sets (Discrete Math)

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From playlist Topology

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From playlist Mathematics

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From playlist Dualities in Topology and Algebra (Online)

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From playlist Discrete Differential Geometry - CMU 15-458/858

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From playlist Topology

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