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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 ...
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.
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.
Retraction (topology) Human physiology. Retracted (phonetics), a sound pronounced to the back of the vocal tract, in linguistics; Retracted tongue root, a position ...
We give a topology by giving it the subspace topology as a subset of (where is the space of paths in which as a function space has the compact-open topology). Then the map E f → B {\displaystyle E_{f}\to B} given by ( a , γ ) ↦ γ ( 1 ) {\displaystyle (a,\gamma )\mapsto \gamma (1)} is a fibration.
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.
Category theory is the language of modern algebra, and has been widely used in the study of algebraic geometry and topology. It has been noted that "the key observation of [10] is that the persistence diagram produced by [8] depends only on the algebraic structure carried by this diagram."