Meta tags:

Headings (most frequently used words):

edit, homotopy, and, theory, concepts, navigation, spaces, tools, contents, key, theorems, obstruction, characteristic, class, localization, completion, of, space, specific, theories, hypothesis, abstract, see, also, references, further, reading, external, links, menu, maps, cofibration, fibration, classifying, operations, spectrum, generalized, cohomology, model, categories, simplicial, personal, namespaces, views, search, contribute, print, export, languages,

Text of the page (most frequently used words):
the (48), #homotopy (42), displaystyle (37), #theory (28), and (26), edit (22), space (20), map (13), that (11), with (10), spaces (10), wikipedia (8), may (8), this (8), spectrum (8), one (8), from (7), algebraic (7), can (7), are (7), theorem (7), cohomology (7), such (7), pointed (7), maps (7), given (7), article (6), main (6), topology (6), called (6), fibration (6), categories (5), articles (5), org (5), category (5), topological (5), example (5), generalized (5), classifying (5), group (5), cofibration (5), you (4), page (4), links (4), help (4), 2020 (4), section (4), abstract (4), mathematics (4), also (4), concepts (4), basic (4), hypothesis (4), there (4), for (4), notion (4), principal (4), bundle (4), about (3), navigation (3), https (3), further (3), see (3), simplicial (3), model (3), adding (3), needs (3), expansion (3), sequence (3), specific (3), theories (3), localization (3), characteristic (3), class (3), higher (3), groups (3), abelian (3), functor (3), where (3), bundles (3), classes (3), free (3), fundamental (3), basepoints (3), which (3), view (2), contact (2), privacy (2), policy (2), terms (2), using (2), 2022 (2), pdf (2), wikidata (2), upload (2), file (2), changes (2), tools (2), contents (2), search (2), talk (2), not (2), expanded (2), short (2), description (2), retrieved (2), homotopy_theory (2), ncatlab (2), nlab (2), external (2), http (2), reading (2), brown (2), isbn (2), whitehead (2), references (2), completion (2), obstruction (2), universal (2), excision (2), key (2), theorems (2), sphere (2), above (2), turns (2), out (2), but (2), other (2), give (2), distinguished (2), mapsto (2), equivalence (2), operations (2), sense (2), exists (2), covering (2), then (2), circ (2), some (2), work (2), often (2), instead (2), groupoid (2), let (2), preserve (2), continuous (2), function (2), works (2), basepoint (2), extra (2), constraints (2), study (2), jump (2), web (2), sep (2), cookie, statement, statistics, developers, mobile, disclaimers, text, available, under, additional, apply, site, agree, registered, trademark, non, profit, organization, wikimedia, foundation, inc, use, creative, commons, attribution, sharealike, license, was, last, edited, august, utc, português, polski, français, العربية, languages, printable, version, download, print, export, item, cite, information, permanent, link, special, pages, related, what, here, recent, community, portal, learn, contribute, donate, random, current, events, more, history, read, views, english, namespaces, log, create, account, contributions, logged, personal, menu, hidden, small, message, boxes, all, different, index, php, title, oldid, 1104063773, show, lectures, martin, frankland, math, 527, spring, 2013, files, cisinski, notes, ronald, 2006, booksurge, llc, 4196, 2722, groupoids, 1978, graduate, texts, vol, 3rd, new, york, berlin, springer, verlag, xxi, 744, 2011, september, 0516508, 978, 387, 90336, elements, george, william, concise, course, pursuing, stacks, type, highly, structured, ring, cofiber, fiber, questions, foundations, nature, asks, whether, something, fundamentally, equivariant, adic, rational, chromatic, stable, simple, several, torsion, postnikov, tower, coefficient, shows, instance, eckmann, hilton, argument, dold, kan, correspondence, landweber, exact, corollary, freudenthal, suspension, seifert, van, kampen, cdots, equation, leads, satisfies, axioms, generalizing, ordinary, always, represented, structure, words, representable, contravariant, eilenberg, maclane, operatorname, idea, classifies, pushed, might, try, classify, mathbb, guarantees, existence, representability, pulling, back, along, refers, isomorphism, left, hand, side, set, each, serre, dual, fact, generalization, projection, action, transitive, extends, loose, analog, defining, diagram, most, since, many, only, complexes, implicit, pair, algebra, injective, module, prefers, variants, definition, points, paths, morphisms, objects, denote, unit, interval, family, indexed, must, when, required, shown, based, particular, geq, integer, relation, times, point, preserves, sends, domain, codomain, contrast, needn, same, vein, possibly, word, denotes, order, avoid, rarely, arbitrary, requires, meet, being, complex, hausdorff, compactly, generated, pathologies, systematic, situations, come, between, them, originated, topic, nowadays, studied, independent, discipline, besides, has, been, used, areas, specifically, geometry, homotopies, branch, encyclopedia, wayback, machine, archive, 20220927213355, wiki, timestamps, capture, fail, success, 2023, 2021, oct, aug, 2004, 2026, 111, captures,

Text of the page (random words):
homotopy theory wikipedia 111 captures 12 may 2004 10 sep 2026 aug sep oct 27 2021 2022 2023 success fail about this capture timestamps the wayback machine http web archive org web 20220927213355 https en wikipedia org wiki homotopy_theory homotopy theory from wikipedia the free encyclopedia jump to navigation jump to search branch of mathematics in mathematics homotopy theory is a systematic study of situations in which maps can come with homotopies between them it originated as a topic in algebraic topology but nowadays is studied as an independent discipline besides algebraic topology the theory has also been used in other areas of mathematics such as algebraic geometry e g a 1 homotopy theory and category theory specifically the study of higher categories contents 1 concepts 1 1 spaces and maps 1 2 homotopy 1 3 cofibration and fibration 1 4 classifying spaces and homotopy operations 1 5 spectrum and generalized cohomology 2 key theorems 3 obstruction theory and characteristic class 4 localization and completion of a space 5 specific theories 6 homotopy hypothesis 7 abstract homotopy theory 7 1 concepts 7 2 model categories 7 3 simplicial homotopy theory 8 see also 9 references 10 further reading 11 external links concepts edit spaces and maps edit in homotopy theory and algebraic topology the word space denotes a topological space in order to avoid pathologies one rarely works with arbitrary spaces instead one requires spaces to meet extra constraints such as being compactly generated or hausdorff or a cw complex in the same vein as above a map is a continuous function possibly with some extra constraints often one works with a pointed space that is a space with a distinguished point called a basepoint a pointed map is then a map which preserves basepoints that is it sends the basepoint of the domain to that of the codomain in contrast a free map is one which needn t preserve basepoints homotopy edit main article homotopy let i denote the unit interval a family of maps indexed by i h t x y displaystyle h_ t x to y is called a homotopy from h 0 displaystyle h_ 0 to h 1 displaystyle h_ 1 if h i x y t x h t x displaystyle h i times x to y t x mapsto h_ t x is a map e g it must be a continuous function when x y are pointed spaces the h t displaystyle h_ t are required to preserve the basepoints a homotopy can be shown to be an equivalence relation given a pointed space x and an integer n 1 displaystyle n geq 1 let π n x s n x displaystyle pi _ n x s n x _ be the homotopy classes of based maps s n x displaystyle s n to x from a pointed n sphere s n displaystyle s n to x as it turns out π n x displaystyle pi _ n x are groups in particular π 1 x displaystyle pi _ 1 x is called the fundamental group of x if one prefers to work with a space instead of a pointed space there is the notion of a fundamental groupoid and higher variants by definition the fundamental groupoid of a space x is the category where the objects are the points of x and the morphisms are paths cofibration and fibration edit a map f a x displaystyle f a to x is called a cofibration if given 1 a map h 0 x z displaystyle h_ 0 x to z and 2 a homotopy g t a z displaystyle g_ t a to z there exists a homotopy h t x z displaystyle h_ t x to z that extends h 0 displaystyle h_ 0 and such that h t f g t displaystyle h_ t circ f g_ t to some loose sense it is an analog of the defining diagram of an injective module in abstract algebra the most basic example is a cw pair x a displaystyle x a since many work only with cw complexes the notion of a cofibration is often implicit a fibration in the sense of serre is the dual notion of a cofibration that is a map p x b displaystyle p x to b is a fibration if given 1 a map z x displaystyle z to x and 2 a homotopy g t z b displaystyle g_ t z to b there exists a homotopy h t z x displaystyle h_ t z to x such that h 0 displaystyle h_ 0 is the given one and p h t g t displaystyle p circ h_ t g_ t a basic example is a covering map in fact a fibration is a generalization of a covering map if e displaystyle e is a principal g bundle that is a space with a free and transitive topological group action of a topological group then the projection map p e x displaystyle p e to x is an example of a fibration classifying spaces and homotopy operations edit given a topological group g the classifying space for principal g bundles the up to equivalence is a space b g displaystyle bg such that for each space x x b g displaystyle x bg principal g bundle on x f f e g displaystyle f mapsto f eg where the left hand side is the set of homotopy classes of maps x b g displaystyle x to bg refers isomorphism of bundles and is given by pulling back the distinguished bundle e g displaystyle eg on b g displaystyle bg called universal bundle along a map x b g displaystyle x to bg brown s representability theorem guarantees the existence of classifying spaces spectrum and generalized cohomology edit main articles spectrum algebraic topology and generalized cohomology the idea that a classifying space classifies principal bundles can be pushed further for example one might try to classify cohomology classes given an abelian group a such as z displaystyle mathbb z x k a n h n x a displaystyle x k a n operatorname h n x a where k a n displaystyle k a n is the eilenberg maclane space the above equation leads to the notion of a generalized cohomology theory i e a contravariant functor from the category of spaces to the category of abelian groups that satisfies the axioms generalizing ordinary cohomology theory as it turns out such a functor may not be representable by a space but it can always be represented by a sequence of pointed spaces with structure maps called a spectrum in other words to give a generalized cohomology theory is to give a spectrum a basic example of a spectrum is a sphere spectrum s 0 s 1 s 2 displaystyle s 0 to s 1 to s 2 to cdots key theorems edit seifert van kampen theorem homotopy excision theorem freudenthal suspension theorem a corollary of the excision theorem landweber exact functor theorem dold kan correspondence eckmann hilton argument this shows for instance higher homotopy groups are abelian universal coefficient theorem obstruction theory and characteristic class edit this section needs expansion you can help by adding to it may 2020 see also characteristic class postnikov tower whitehead torsion localization and completion of a space edit this section needs expansion you can help by adding to it may 2020 main article localization of a topological space specific theories edit there are several specific theories simple homotopy theory stable homotopy theory chromatic homotopy theory rational homotopy theory p adic homotopy theory equivariant homotopy theory homotopy hypothesis edit main article homotopy hypothesis one of the basic questions in the foundations of homotopy theory is the nature of a space the homotopy hypothesis asks whether a space is something fundamentally algebraic abstract homotopy theory edit concepts edit fiber sequence cofiber sequence model categories edit this section needs expansion you can help by adding to it may 2020 main article model category simplicial homotopy theory edit simplicial homotopy see also edit highly structured ring spectrum homotopy type theory pursuing stacks references edit may j a concise course in algebraic topology george william whitehead 1978 elements of homotopy theory graduate texts in mathematics vol 61 3rd ed new york berlin springer verlag pp xxi 744 isbn 978 0 387 90336 1 mr 0516508 retrieved september 6 2011 ronald brown topology and groupoids 2006 booksurge llc isbn 1 4196 2722 8 further reading edit cisinski s notes http ncatlab org nlab files abstract homotopy pdf math 527 homotopy theory spring 2013 section f1 lectures by martin frankland external links edit https ncatlab org nlab show homotopy theory retrieved from https en wikipedia org w index php title homotopy_theory oldid 1104063773 categories homotopy theory hidden categories articles with short description short description is different from wikidata articles to be expanded from may 2020 all articles to be expanded articles using small message boxes navigation menu personal tools not logged in talk contributions create account log in namespaces article talk english views read edit view history more search navigation main page contents current events random article about wikipedia contact us donate contribute help learn to edit community portal recent changes upload file tools what links here related changes upload file special pages permanent link page information cite this page wikidata item print export download as pdf printable version languages العربية français polski português edit links this page was last edited on 12 august 2022 at 11 25 utc text is available under the creative commons attribution sharealike license 3 0 additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia mobile view developers statistics cookie statement