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In algebraic topology, a simplicial homotopy [1] pg 23 is an analog of a homotopy between topological spaces for simplicial sets.If ,: are maps between simplicial sets, a simplicial homotopy from f to g is a map
In mathematics, simple homotopy theory is a homotopy theory (a branch of algebraic topology) that concerns with the simple-homotopy type of a space. It was originated by Whitehead in his 1950 paper "Simple homotopy types".
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.
The simplicial homology groups H n (X) of a simplicial complex X are defined using the simplicial chain complex C(X), with C n (X) the free abelian group generated by the n-simplices of X. See simplicial homology for details.
Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex .
A CW complex is a space that has a filtration whose union is and such that . is a discrete space, called the set of 0-cells (vertices) in .; Each is obtained by attaching several n-disks, n-cells, to via maps ; i.e., the boundary of an n-disk is identified with the image of in .
In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves ...
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 ...