Editing Homotopy (section)

{{short description|Continuous deformation between two continuous functions}}
{{About|topology|chemistry|Homotopic groups}}
[[File:HomotopySmall.gif|thumb|The two dashed [[Path (topology)|paths]] shown above are homotopic relative to their endpoints. The animation represents one possible homotopy.]]
In [[topology]], two [[continuous function (topology)|continuous function]]s from one [[topological space]] to another are called '''homotopic''' (from {{langx|grc|ὁμός}} {{Transliteration|grc|homós}} {{gloss|same, similar}} and {{lang|grc|τόπος}} {{Transliteration|grc|tópos}} {{gloss|place}}) if one can be "continuously deformed" into the other, such a deformation being called a '''homotopy''' ({{IPAc-en|h|ə|ˈ|m|ɒ|t|ə|p|iː}}<ref>{{cite web |url = https://www.dictionary.com/browse/homotopy |title = Homotopy Definition & Meaning |access-date = 22 April 2022 }}</ref> {{Respell|hə|MOT|ə|pee}}; {{IPAc-en|ˈ|h|oʊ|m|oʊ|ˌ|t|oʊ|p|iː}}<ref>{{cite web |url = https://www.youtube.com/watch?v=Ft8R3-kPDdk |title = Homotopy Type Theory Discussed - Computerphile |website = [[YouTube]] |date = 13 October 2017 |access-date = 22 April 2022 }}</ref> {{Respell|HOH|moh|toh|pee}}) between the two functions. A notable use of homotopy is the definition of [[homotopy groups]] and [[cohomotopy groups]], important [[invariant (mathematics)|invariant]]s in [[algebraic topology]].<ref>{{Cite web|url=https://www.britannica.com/science/homotopy|title=Homotopy {{!}} mathematics|website=Encyclopedia Britannica|language=en|access-date=2019-08-17}}</ref>

In practice, there are technical difficulties in using homotopies with certain spaces.  Algebraic topologists work with [[compactly generated space]]s, [[CW complex]]es, or [[Spectrum (homotopy theory)|spectra]].