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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/10638/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This page is about homotopy as a transformation. For homotopy sets in <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a>, see <a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy (as an operation)</a>.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="type_theory">Type theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a></strong> <a class="existingWikiWord" href="/nlab/show/metalanguage">metalanguage</a>, <a class="existingWikiWord" href="/nlab/show/practical+foundations">practical foundations</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/judgement">judgement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypothetical+judgement">hypothetical judgement</a>, <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/antecedents">antecedents</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/consequent">consequent</a>, <a class="existingWikiWord" href="/nlab/show/succedents">succedents</a></li> </ul> </li> </ul> <ol> <li><a class="existingWikiWord" href="/nlab/show/type+formation+rule">type formation rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+introduction+rule">term introduction rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+elimination+rule">term elimination rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/computation+rule">computation rule</a></li> </ol> <p><strong><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent</a>, <a class="existingWikiWord" href="/nlab/show/intensional+type+theory">intensional</a>, <a class="existingWikiWord" href="/nlab/show/observational+type+theory">observational type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/calculus+of+constructions">calculus of constructions</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/syntax">syntax</a></strong> <a class="existingWikiWord" href="/nlab/show/object+language">object language</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory">theory</a>, <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>/<a class="existingWikiWord" href="/nlab/show/type">type</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definition">definition</a>/<a class="existingWikiWord" href="/nlab/show/proof">proof</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a> (<a class="existingWikiWord" href="/nlab/show/proofs+as+programs">proofs as programs</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorem">theorem</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/computational+trinitarianism">computational trinitarianism</a></strong> = <br /> <strong><a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/programs+as+proofs">programs as proofs</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation type theory/category theory</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/logic">logic</a></th><th><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> (<a class="existingWikiWord" href="/nlab/show/internal+logic+of+set+theory">internal logic</a> of)</th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object">object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/predicate">predicate</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/family+of+sets">family of sets</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/display+morphism">display morphism</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof">proof</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/element">element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/term">term</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+rule">cut rule</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/classifying+morphisms">classifying morphisms</a> / <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <a class="existingWikiWord" href="/nlab/show/display+maps">display maps</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/substitution">substitution</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/introduction+rule">introduction rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/counit">counit</a> for hom-tensor adjunction</td><td style="text-align: left;">lambda</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elimination+rule">elimination rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> for hom-tensor adjunction</td><td style="text-align: left;">application</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+elimination">cut elimination</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">one of the <a class="existingWikiWord" href="/nlab/show/zigzag+identities">zigzag identities</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/beta+reduction">beta reduction</a></td></tr> <tr><td style="text-align: left;">identity elimination for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">the other <a class="existingWikiWord" href="/nlab/show/zigzag+identity">zigzag identity</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/eta+conversion">eta conversion</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/true">true</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/singleton">singleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-2%29-truncated+object">(-2)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+0">h-level 0</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/false">false</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>, <a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncated+object">(-1)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>, <a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product">product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product+type">product type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> (into <a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (into <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> (into <a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/negation">negation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> into <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> into <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> into <a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subsingletons">subsingletons</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subterminal+objects">subterminal objects</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> (of family of <a class="existingWikiWord" href="/nlab/show/h-propositions">h-propositions</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bijection+set">bijection set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+of+isomorphisms">object of isomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+type">equivalence type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+set">support set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+object">support object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/propositional+truncation">propositional truncation</a>/<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-image">n-image</a> of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> into <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/n-truncation">n-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equality">equality</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/diagonal+function">diagonal function</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+subset">diagonal subset</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+relation">diagonal relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>/<a class="existingWikiWord" href="/nlab/show/path+type">path type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/completely+presented+set">completely presented set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>/<a class="existingWikiWord" href="/nlab/show/0-truncated+object">0-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+2">h-level 2</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/set">set</a>/<a class="existingWikiWord" href="/nlab/show/h-set">h-set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> with <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">internal 0-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a>/<a class="existingWikiWord" href="/nlab/show/setoid">setoid</a> with its <a class="existingWikiWord" href="/nlab/show/pseudo-equivalence+relation">pseudo-equivalence relation</a> an actual <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a>/<a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+type">quotient type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a>, <a class="existingWikiWord" href="/nlab/show/W-type">W-type</a>, <a class="existingWikiWord" href="/nlab/show/M-type">M-type</a></td></tr> <tr><td style="text-align: left;">higher <a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a></td></tr> <tr><td style="text-align: left;">-</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a> <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+inductive+type">quotient inductive type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinduction">coinduction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/limit">limit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinductive+type">coinductive type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/preset">preset</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a> without <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+of+propositions">type of propositions</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+of+discourse">domain of discourse</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universe">universe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modality">modality</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/closure+operator">closure operator</a>, (<a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a>) <a class="existingWikiWord" href="/nlab/show/monad">monad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal type theory</a>, <a class="existingWikiWord" href="/nlab/show/monad+%28in+computer+science%29">monad (in computer science)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></td><td style="text-align: left;"></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a>) <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a>/<a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof+net">proof net</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a></td></tr> <tr><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/contraction+rule">contraction rule</a></td><td style="text-align: left;"></td><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/synthetic+mathematics">synthetic mathematics</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+specific+embedded+programming+language">domain specific embedded programming language</a></td></tr> </tbody></table> </div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+levels">homotopy levels</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-type+theory">2-type theory</a>, <a class="existingWikiWord" href="/michaelshulman/show/2-categorical+logic">2-categorical logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory+-+contents">homotopy type theory - contents</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a>, <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>, <a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+homotopy+type+theory">directed homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HoTT+methods+for+homotopy+theorists">HoTT methods for homotopy theorists</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/semantics">semantics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>, <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/display+map">display map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+a+topos">internal logic of a topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mitchell-Benabou+language">Mitchell-Benabou language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kripke-Joyal+semantics">Kripke-Joyal semantics</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/type-theoretic+model+category">type-theoretic model category</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/type+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#in_topological_spaces'>In topological spaces</a></li> <li><a href='#in_enriched_categories'>In enriched categories</a></li> <li><a href='#in_model_categories'>In model categories</a></li> <li><a href='#in_cofibration_categories'>In (co-)fibration categories</a></li> <li><a href='#in_dependent_type_theory'>In dependent type theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In many <a class="existingWikiWord" href="/nlab/show/category">categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in which one does <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, there is a notion of <em>homotopy</em> between <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>, which is closely related to the <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>: a homotopy between two morphisms is a way in which they are equivalent.</p> <p>If we regard such a category as a presentation of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">category</a>, then homotopies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∼</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f\sim g</annotation></semantics></math> present the 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f\Rightarrow g</annotation></semantics></math> in the resulting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category.</p> <h2 id="definition">Definition</h2> <h3 id="in_topological_spaces">In topological spaces</h3> <div class="num_defn" id="LeftHomotopy"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g\colon X \longrightarrow Y</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math>, then a <strong>left homotopy</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>f</mi><mspace width="thinmathspace"></mspace><msub><mo>⇒</mo> <mi>L</mi></msub><mspace width="thinmathspace"></mspace><mi>g</mi></mrow><annotation encoding="application/x-tex"> \eta \colon f \,\Rightarrow_L\, g </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mi>I</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; X \times I \longrightarrow Y </annotation></semantics></math></div> <p>out of the standard <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>: the <a class="existingWikiWord" href="/nlab/show/product+space">product space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>≔</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I \coloneqq [0,1]</annotation></semantics></math>, such that this fits into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div style="float:right;margin:0 10px 10px 0;"> <img src="http://www.ncatlab.org/nlab/files/AHomotopy.jpg" width="400" /> </div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>g</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,. </annotation></semantics></math></div> <p>(graphics grabbed from J. Tauber <a href="http://jtauber.com/blog/2005/07/01/path_homotopy/">here</a>)</p> </div> <div class="num_example" id="PathsAsLeftHomotopyBetweenPoints"> <h6 id="example">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x,y \in X</annotation></semantics></math> be two of its points, regarded as functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x,y \colon \ast \longrightarrow X</annotation></semantics></math> from the point to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Then a left homotopy, def. <a class="maruku-ref" href="#LeftHomotopy"></a>, between these two functions is a commuting diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>δ</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>x</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>δ</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>y</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \ast \\ {}^{\mathllap{\delta_0}}\downarrow & \searrow^{\mathrlap{x}} \\ I &\stackrel{\eta}{\longrightarrow}& X \\ {}^{\mathllap{\delta_1}}\uparrow & \nearrow_{\mathrlap{y}} \\ \ast } \,. </annotation></semantics></math></div> <p>This is simply a continuous path in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> whose endpoints are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>.</p> </div> <h3 id="in_enriched_categories">In enriched categories</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, then a <strong>homotopy</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> between maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>⇉</mo><mspace width="thinmathspace"></mspace><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g:X\,\rightrightarrows \,Y</annotation></semantics></math> is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H:[0,1] \to C(X,Y)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">H(0)=f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">H(1)=g</annotation></semantics></math>. In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> itself this is the classical notion.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/copower">copowers</a>, then an equivalent definition is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⊙</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">[0,1]\odot X\to Y</annotation></semantics></math>, while if it has <a class="existingWikiWord" href="/nlab/show/power">powers</a>, an equivalent definition is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>⋔</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\to \pitchfork([0,1],Y)</annotation></semantics></math>.</p> <p>There is a similar definition in a <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a>, replacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> with the 1-simplex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Delta^1</annotation></semantics></math>, with the caveat that in this case not all <em>simplicial homotopies</em> need be composable even if they match correctly. (This depends on whether or not all (2,1)-<a class="existingWikiWord" href="/nlab/show/horn">horn</a>s in the simplicial set, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,Y)</annotation></semantics></math>, have fillers.) Likewise in a <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a> we can use the “chain complex interval” to get a notion of <em>chain homotopy</em>.</p> <h3 id="in_model_categories">In model categories</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, it has an intrinsic notion of homotopy determined by its factorizations. For more on the following see at <em><a class="existingWikiWord" href="/nlab/show/homotopy+in+a+model+category">homotopy in a model category</a></em>.</p> <div class="num_defn" id="PathAndCylinderObjectsInAModelCategory"> <h6 id="definition_3">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/object">object</a>.</p> <ul> <li>A <strong><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a factorization of the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\nabla_X \colon X \to X \times X</annotation></semantics></math> as</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mi>i</mi></munderover><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla_X \;\colon\; X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X \,. </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\to Path(X)</annotation></semantics></math> is a weak equivalence. This is called a <strong>good path object</strong> if in addition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Path(X) \to X \times X</annotation></semantics></math> is a fibration.</p> <ul> <li>A <strong><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(X)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a factorization of the <a class="existingWikiWord" href="/nlab/show/codiagonal">codiagonal</a> (or “fold map”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>⊔</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Delta_X: X \sqcup X \to X</annotation></semantics></math> as</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mi>p</mi><mrow><mo>∈</mo><mi>W</mi></mrow></munderover><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta_X \;\colon\; X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X \,. </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Cyl(X) \to X</annotation></semantics></math> is a weak equivalence. This is called a <strong>good cylinder object</strong> if in addition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi><mo>→</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sqcup X \to Cyl(X)</annotation></semantics></math> is a cofibration.</p> </div> <div class="num_remark" id="RemarkOnChoicesOfNonGoodPathAndCylinderObjects"> <h6 id="remark">Remark</h6> <p>By the factorization axioms every object in a model category has both a good path object and as well as a good cylinder object according to def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>. But in some situations one is genuinely interested in using non-good such objects.</p> <p>For instance in the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>, the obvious object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X\times [0,1]</annotation></semantics></math> is a cylinder object, but not a good cylinder unless <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> itself is cofibrant (a <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a> in this case).</p> <p>More generally, the path object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Path</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Path(X)</annotation></semantics></math> of def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a> is analogous to the <a class="existingWikiWord" href="/nlab/show/powering">powering</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋔</mo><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pitchfork(I,X)</annotation></semantics></math> with an <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> and the cylinder object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cyl(X)</annotation></semantics></math> is analogous to the <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊙</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">I\odot X</annotation></semantics></math> with an interval object. In fact, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is fibrant/cofibrant, then these powers and copowers are in fact examples of (good) path and cylinder objects if the <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a> is sufficiently good.</p> </div> <div class="num_defn" id="LeftAndRightHomotopyInAModelCategory"> <h6 id="definition_4">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g \colon X \longrightarrow Y</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> in a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>.</p> <ul> <li>A <strong>left homotopy</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>f</mi><msub><mo>⇒</mo> <mi>L</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">\eta \colon f \Rightarrow_L g</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\eta \colon Cyl(X) \longrightarrow Y</annotation></semantics></math> from a <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>, such that it makes this <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commute</a>:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cyl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟵</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>η</mi></mpadded></msup></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>g</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& Cyl(X) &\longleftarrow& X \\ & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}} \\ && Y } \,. </annotation></semantics></math></div> <ul> <li>A <strong>right homotopy</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>f</mi><msub><mo>⇒</mo> <mi>R</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">\eta \colon f \Rightarrow_R g</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta \colon X \to Path(Y)</annotation></semantics></math> to some <a class="existingWikiWord" href="/nlab/show/path+object">path object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a>, such that this <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a>:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo>↙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>η</mi></mpadded></msup></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>⟵</mo></mtd> <mtd><mi>Path</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X \\ & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}} \\ Y &\longleftarrow& Path(Y) &\longrightarrow& Y } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>By remark <a class="maruku-ref" href="#RemarkOnChoicesOfNonGoodPathAndCylinderObjects"></a> it follows that in a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a>, any enriched homotopy between maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\to Y</annotation></semantics></math> is a left homotopy if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is cofibrant and a right homotopy if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is fibrant. Similar remarks hold for other enrichments.</p> </div> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/homotopy+in+a+model+category">homotopy in a model category</a></em>.</p> <h3 id="in_cofibration_categories">In (co-)fibration categories</h3> <p>Clearly the concept of left homotopy in def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a> only needs part of the model category axioms and thus makes sense more generally in suitable <a class="existingWikiWord" href="/nlab/show/cofibration+categories">cofibration categories</a>. Dually, the concept of path objects in def. <a class="maruku-ref" href="#PathAndCylinderObjectsInAModelCategory"></a> makes sense more generally in suitable <a class="existingWikiWord" href="/nlab/show/fibration+categories">fibration categories</a> such as <a class="existingWikiWord" href="/nlab/show/categories+of+fibrant+objects">categories of fibrant objects</a> in the sense of Brown.</p> <p>Likewise if there is a <a class="existingWikiWord" href="/nlab/show/cylinder+functor">cylinder functor</a>, one gets functorially defined <a class="existingWikiWord" href="/nlab/show/cylinder+objects">cylinder objects</a>, etc.</p> <h3 id="in_dependent_type_theory">In dependent type theory</h3> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> be a type and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/type+family">type family</a> indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f,g:\prod_{x:A} P(x)</annotation></semantics></math> be two elements of a <a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> of a <a class="existingWikiWord" href="/nlab/show/type+family">type family</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>. The <strong>type of homotopies</strong> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is the type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∼</mo><mi>g</mi><mo>≡</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \sim g \equiv \prod_{x:A} (f(x) =_{B(x)} g(x))</annotation></semantics></math></div> <p>A <strong>homotopy</strong> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is simply an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>:</mo><mi>f</mi><mo>∼</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">H:f \sim g</annotation></semantics></math>.</p> <p>Note that a homotopy is not the same as an <a class="existingWikiWord" href="/nlab/show/identification">identification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f = g</annotation></semantics></math>. However this can be made so if one assumes <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>.</p> <h2 id="related_concepts">Related concepts</h2> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></th><th><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/homology">homology</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[S^n,-]</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,A]</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(-) \otimes A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/covariant+functor">covariant</a> <a class="existingWikiWord" href="/nlab/show/hom+functor">hom</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant</a> <a class="existingWikiWord" href="/nlab/show/hom+functor">hom</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/end">end</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/end">end</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coend">coend</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(S^n,-)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(-,A)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derived+tensor+product">derived tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo>⊗</mo> <mi>𝕃</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">(-) \otimes^{\mathbb{L}} A</annotation></semantics></math></td></tr> </tbody></table> </div> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+relative+boundary">homotopy relative boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+homotopy">smooth homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isotopy">isotopy</a>, <a class="existingWikiWord" href="/nlab/show/smooth+isotopy">smooth isotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+class">homotopy class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudonatural+transformation">pseudonatural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+analysis+method">homotopy analysis method</a></p> </li> </ul> <h2 id="references">References</h2> <blockquote> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em> and at <em><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></em>.</p> </blockquote> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Anatoly+Fomenko">Anatoly Fomenko</a>, <a class="existingWikiWord" href="/nlab/show/Dmitry+Fuchs">Dmitry Fuchs</a>, §3.1 in: <em>Homotopical Topology</em>, Graduate Texts in Mathematics <strong>273</strong>, Springer (2016) [<a href="https://doi.org/10.1007/978-3-319-23488-5">doi:10.1007/978-3-319-23488-5</a>, <a href="https://www.cimat.mx/~gil/docencia/2020/topologia_diferencial/[Fomenko,Fuchs]Homotopical_Topology(2016).pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Adam+Marsh">Adam Marsh</a>: Fig. 4.2.2 in: <em>Mathematics for Physics: An Illustrated Handbook</em>, World Scientific (2018) [<a href="https://doi.org/10.1142/10816">doi:10.1142/10816</a>, <a href="https://www.mathphysicsbook.com/">book webpage</a>]</p> </li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/computational+topology">computational topology</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marek+Filakovsk%C3%BD">Marek Filakovský</a>, <a class="existingWikiWord" href="/nlab/show/Luk%C3%A1%C5%A1+Vok%C5%99%C3%ADnek">Lukáš Vokřínek</a>, <em>Are two given maps homotopic? An algorithmic viewpoint</em>, Found Comput Math (2019) (<a href="https://arxiv.org/abs/1312.2337">arXiv:1312.2337</a>, <a href="https://doi.org/10.1007/s10208-019-09419-x">doi:10.1007/s10208-019-09419-x</a>)</li> </ul> <p>For homotopies in <a class="existingWikiWord" href="/nlab/show/Martin-L%C3%B6f+dependent+type+theory">Martin-Löf dependent type theory</a>:</p> <ul> <li>Univalent Foundations Project, <a class="existingWikiWord" href="/nlab/show/HoTT+book">Homotopy Type Theory – Univalent Foundations of Mathematics</a> (2013)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 22, 2024 at 07:12:11. 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