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A simplicial 3-complex. In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their n-dimensional counterparts, called simplices, such that all the faces and intersections of the elements are also included in the set (see illustration).
Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. [1] 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.
A simplicial line arrangement (left) and a simple line arrangement (right). In geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons , the cells of the arrangement, line segments and rays , the edges of the arrangement, and ...
A simplicial map : is said to be a simplicial approximation of if and only if each is mapped by onto the support of () in . If such an approximation exists, one can construct a homotopy H {\displaystyle H} transforming f {\displaystyle f} into g {\displaystyle g} by defining it on each simplex; there it always exists, because simplices are ...
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
In topology and combinatorics, it is common to "glue together" simplices to form a simplicial complex. The geometric simplex and simplicial complex should not be confused with the abstract simplicial complex, in which a simplex is simply a finite set and the complex is a family of such sets that is closed under taking subsets.
The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a simplicial set. Briefly speaking, a simplicial set K ∗ {\displaystyle K_{*}} can be described by a collection of sets K i , i ≥ 0 {\displaystyle K_{i},\ i\geq 0} , together with face and degeneracy maps between them satisfying a ...
A simplicial set is called a Kan complex if the map from {}, the one-point simplicial set, is a Kan fibration. In the model category for simplicial sets, { ∗ } {\displaystyle \{*\}} is the terminal object and so a Kan complex is exactly the same as a fibrant object .
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