Section (category theory)

In category theory is meant by a retraction of a morphism which has a right inverse, that is, for which there is a morphism. The term dual is the one retraction of Koretraktion (or weight), that is a morphism that has a left- inverse. The right inverse of a retraction is a Koretraktion and vice versa.

An object of a category is called a retract of an object when it comes to in a morphism and a retraction, ie a morphism, there.

Every retraction is an extreme and even regular epimorphism. Similarly, any Koretraktion extreme and even regular monomorphism and even differential core.

Special Categories

Topological spaces

The concept of retraction has applications in algebraic topology. In the category of topological spaces all extreme monomorphisms and thus all Koretraktionen are topological embeddings. This allows, in the case of topological spaces a different view and definition: A retraction is a continuous left inverse of a topological embedding. Or concretely formulated: A retraction is a continuous mapping of a topological space into itself, so that each element of the image set is a fixed point.

This also allows for a concrete definition of Retrakts: A subspace A of a topological space X is called a retract of X if there is a retraction r for embedding.

A is then retract of X if every continuous map can be continued continuously to a mapping:

  • Is there a retraction, so is continuous extension.
  • A continuation of a continuous map is a retraction.

In a Hausdorff space every retract is finished: Be retract with retraction. Consider now on a converged network. The image network converges ( steadily since ) and is equal to the original network. Since the limit of a network is unique in Hausdorff spaces, thus applies and is complete. In non- Hausdorff spaces this is not true: In non-T ₁ - spaces exist non- completed one-element sets, but which are obviously Retrakte. As an example of a T ₁ - space with non- completed retract consider the kofinite topology: with and is a retraction, the image is not complete.

Deformation retract

A is called a deformation retract when is homotopic relative to A.

Deformationsretraktionen are special homotopy equivalences which generate this equivalence relation.


Elementary example

The following figure is an illustrative example of a retraction in the real numbers:

Brouwer fixed point theorem in the one-dimensional case

The Brouwer fixed point theorem states that any continuous map of a solid sphere has a fixed point in itself. A one-dimensional solid sphere is topologically just saw a closed interval, approximately. If there was a steady, fixed-point- free imaging, there would result by a retraction means ( since the denominator would never disappear ), that is, would retract of his. However, such a retraction can not exist, because the context is preserved under continuous maps.

Closed subspaces of the Baire space

The Baire space is considered: For any closed subspaces (these are always Polish subspaces ) is a retract of. Note that the Baire space is totally disconnected, and therefore the related concept provides no restrictions for Retrakte.

Arrow Category

Let be a category, the corresponding arrow category is then the category of functors from the category with two objects and three morphisms in the category. These arrows are called and can be identified with the morphisms in. An arrow is a retract of an arrow when there is a natural transformation (ie a commuting square) and retraction are so commutes the following diagram:

Set theory

In the category of all sets and functions between them is a morphism (that is a function between two sets ) if and only retraction if it is surjective. This statement is equivalent to the axiom of choice in set theory. Similarly, a morphism if and only a Koretraktion if it is injective and there is a morphism in the opposite direction. However, this statement does not require the axiom of choice. From these statements it follows that in every concrete category, the retractions surjective and the Koretraktionen must be injective, which generally does not apply to general epi- and monomorphisms, which coincide in the category of sets with the retractions or Koretraktionen.