Homotopy theory MOC

Homotopy of maps

A homotopy is a continuous transformation from one continuous map into another. Let . Then a homotopy from to is a continuous map such that and . The maps are thereby said to be homotopic, denoted with homotopy It is useful to have , whereby we can say and .

The homotopy relation is a congruence relation on .

Proof

Clearly is homotopic to itself via , so is reflexive. If is a homotopy from to then is a homotopy from to , so is symmetric. If is a homotopy from to and is a homotopy from to , then

is a homotopy from to , so is transitive. Therefore is an equivalence relation. To show is a congruence relation, let with and with . Then , and similarly . Thus , as required.

Homotopy class

The congruence classes of homotopic maps are called homotopy classes of maps, and form the morphisms in the Naïve homotopy category , which is a Quotient category .

Other kinds of topological homotopy

• Homotopy of paths
• Pointed homotopy of maps

Further terminology

• A map is said to be null-homotopic iff it is homotopic to a Constant map.

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