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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".
Let S and T be simplicial complexes. A simplicial map f from S to T is a function from the vertex set of S to the vertex set of T such that the image of each simplex in S (viewed as a set of vertices) is a simplex in T. A simplicial map f: S → T determines a homomorphism of homology groups H k (S) → H k (T) for each integer k.
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 1967 for his studies on algebraic topology he received a Guggenheim Fellowship [1] [2] and in 1972 the Leroy P. Steele Prize for his paper Simplicial homotopy theory. [3] Curtis died in Seattle, Washington on April 2, 2024, at the age of 91. [4]
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 ...
In mathematics, particularly the area of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence.Two CW-complexes are simple-homotopy equivalent if they are related by a sequence of collapses and expansions (inverses of collapses), and a homotopy equivalence is a simple homotopy equivalence if it is homotopic to such a map.
The older definition of the homotopy category hTop, called the naive homotopy category [1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f : X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other.