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In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. [1] The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a ...
In mathematics, specifically algebraic topology, the mapping cylinder [1] of a continuous function between topological spaces and is the quotient = (([,])) / where the denotes the disjoint union, and ~ is the equivalence relation generated by
a retraction if a right inverse of f exists, i.e. if there exists a morphism g : b → a with f ∘ g = 1 b. a section if a left inverse of f exists, i.e. if there exists a morphism g : b → a with g ∘ f = 1 a. Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent:
The concept of a retraction in category theory comes from the essentially similar notion of a retraction in topology: : where is a subspace of is a retraction in the topological sense, if it's a retraction of the inclusion map : in the category theory sense.
There is a cofibration (A, X), if and only if there is a retraction from to () ({}), since this is the pushout and thus induces maps to every space sensible in the diagram. Similar equivalences can be stated for deformation-retract pairs, and for neighborhood deformation-retract pairs.
The homotopy extension property is depicted in the following diagram If the above diagram (without the dashed map) commutes (this is equivalent to the conditions above), then pair (X,A) has the homotopy extension property if there exists a map f ~ ∙ {\displaystyle {\tilde {f}}_{\bullet }} which makes the diagram commute.
The normal complement, specifically, is the kernel of the retraction. Every direct factor is a retract. [1] Conversely, any retract which is a normal subgroup is a direct factor. [5] Every retract has the congruence extension property. Every regular factor, and in particular, every free factor, is a retract.
In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming a simplex for every finite set of points that has diameter at most δ.