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 whose objects are topological spaces and whose morphisms are homotopy classes of maps. In other words, homotopypreserving functors induce a well-defined functor HoTop → C, making homotopy-invariant constructions compatible with the categorical structure.

Examples of homotopypreserving functors include the identity functor on Top, the product with a fixed space X (− × X), and loop-space constructions that preserve homotopies of based maps. Classical homology and homotopy functors, such as singular homology Hn(−) and the fundamental groupoid π1(−), are typically homotopy-preserving in the appropriate sense, since homotopic maps induce the same maps on homology or have compatible induced morphisms on π1.

Not all functors are homotopy-preserving; a functor that does not respect homotopy information may fail to descend to HoTop. Understanding whether a functor is homotopypreserving helps determine its behavior on spaces up to homotopy equivalence. See also: homotopy category, homotopy invariance, derived functors.