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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: ( ) ∨ ( ) ∨ ( ) .
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 ...
Google Translate is a multilingual neural machine translation service developed by Google to translate text, documents and websites from one language into another. It offers a website interface, a mobile app for Android and iOS, as well as an API that helps developers build browser extensions and software applications. [3]
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 ...
DeepL Translator is a neural machine translation service that was launched in August 2017 and is owned by Cologne-based DeepL SE.The translating system was first developed within Linguee and launched as entity DeepL.
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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.