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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).
The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is the join of two points: ( ) ∨ ( ) = 2 ⋅ ( ) . A general 2-simplex (scalene triangle) is the join of three points: ( ) ∨ ( ) ∨ ( ) .
This can be achieved in a greedy way by iteratively removing from the simplicial complex the highest order simplexes until the simplicial complex is empty. We then need to label each vertex from 1 to | V | {\displaystyle \left\vert V\right\vert } and associate each simplex with its corresponding "word", that is the ordered list of its vertices ...
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 ...
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 map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of a simplex always span a simplex. [1] Simplicial maps can be used to approximate continuous functions between topological spaces that can be triangulated ; this is formalized by the simplicial ...
The U.S. State Department has removed a statement on its website that it does not support Taiwan independence, among changes that the island's government praised on Sunday as supporting Taiwan.
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 .