homotopypreserving
Homotopypreserving is a term used to describe a property of a functor or construction in topology and related areas, indicating that it respects homotopy relations. In precise terms, a functor F between categories of spaces (and possibly additional structure, such as pointed spaces or chain complexes) is called homotopypreserving if whenever two continuous maps f, g: X → Y are homotopic, their images under F are also homotopic: F(f) ≃ F(g). If the target category has its own notion of homotopy between morphisms (for example, chain homotopy in chain complexes), the requirement is F(f) and F(g) are related by that notion in the target.
This property has a standard consequence: the functor F factors through the homotopy category HoTop, the category
Examples of homotopypreserving functors include the identity functor on Top, the product with a fixed space
Not all functors are homotopy-preserving; a functor that does not respect homotopy information may fail to