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href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <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>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> <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="modalities_closure_and_reflection">Modalities, Closure and Reflection</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal type theory</a></strong>, <a class="existingWikiWord" href="/nlab/show/modal+logic">modal logic</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/closure+operator">closure operator</a></strong>, <a class="existingWikiWord" href="/nlab/show/universal+closure+operator">universal closure operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modal+type">modal type</a>, <a class="existingWikiWord" href="/nlab/show/local+object">local object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a>, <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Moore+closure">Moore closure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+modality">geometric modality</a>/<a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">Lawvere-Tierney topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S4+modal+logic">S4 modal logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-truncated+object+in+an+%28infinity%2C1%29-category">n-truncation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a></p> </li> </ul> </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='#InTermsOfTruncations'>In terms of truncations</a></li> <li><a href='#in_terms_of_categorical_homotopy_groups'>In terms of categorical homotopy groups</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RecursiveDefinition'>Recursive definition</a></li> <li><a href='#PropertiesGeneral'>General</a></li> <li><a href='#ModelCategoryPresentations'>Model category presentations</a></li> </ul> <li><a href='#truncation'>Truncation</a></li> <ul> <li><a href='#definition_5'>Definition</a></li> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#GeneralPropsTruncation'>General</a></li> <li><a href='#postnikov_tower'>Postnikov tower</a></li> <li><a href='#TruncationHomotopyTypeTheorySyntax'>Homotopy type theory syntax</a></li> </ul> <li><a href='#RelationToHomotopyGroups'>Relation to homotopy groups</a></li> <ul> <li><a href='#observation'>Observation</a></li> </ul> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#truncated_morphisms'>Truncated morphisms</a></li> <ul> <li><a href='#general_3'>General</a></li> <li><a href='#TruncatedMorphismsBetweenGroupoids'>Between groupoids</a></li> <li><a href='#TruncatedMorphismsBetweenStacks'>Between stacks</a></li> </ul> <li><a href='#InInfGrpd'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> and 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></a></li> <ul> <li><a href='#observation_2'>Observation</a></li> </ul> <li><a href='#DiagonalExamples'>Diagonals</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#in_category_theory'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category theory</a></li> <li><a href='#in_homotopy_type_theory'>In homotopy type theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>An <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></strong> is an <a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a>.</p> <p>An <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></strong> is a <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a>: all <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> above degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> are trivial.</p> <p>An <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated object</strong> in a general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> is an object such that all <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">hom-∞-groupoids</a> into it are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated.</p> <p>If an object in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated for any (possibly large) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, then it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated precisely if all its <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups</a> above degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> are trivial.</p> <p>The complementary notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated object is that of an <a class="existingWikiWord" href="/nlab/show/n-connected+object+of+an+%28%E2%88%9E%2C1%29-category">n-connected object of an (∞,1)-category</a>.</p> <h2 id="definition">Definition</h2> <h3 id="InTermsOfTruncations">In terms of truncations</h3> <div class="num_defn" id="nTruncatedInfinityGroupoid"> <h6 id="definition_2">Definition</h6> <p><strong>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid)</strong></p> <p>An <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">A \in \infty Grpd</annotation></semantics></math> is <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> if it is an <a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a>:</p> <p>Precisely: in the model of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid"> ∞-groupoids</a> given by <a class="existingWikiWord" href="/nlab/show/Kan+complex"> Kan complexes</a> <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> is <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated</strong> if the <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group"> simplicial homotopy groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_k(A,x)</annotation></semantics></math> are trivial for all <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 all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \gt n</annotation></semantics></math>.</p> </div> <p>It makes sense for the following to adopt the convention that <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> is called.</p> <ul> <li> <p><em><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><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1)</annotation></semantics></math>-truncated</em> if it is empty or contractible – this is a <a class="existingWikiWord" href="/nlab/show/%28-1%29-groupoid">(-1)-groupoid</a>.</p> </li> <li> <p><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><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-2)</annotation></semantics></math>-truncated if it is non-empty and contractible – this is a <a class="existingWikiWord" href="/nlab/show/%28-2%29-groupoid">(-2)-groupoid</a>.</p> </li> </ul> <p>(following <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, p. 6</a>).</p> <p>To generalize this, let now <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> be an arbitrary <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X,A</annotation></semantics></math> objects 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> write <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>A</mi><mo stretchy="false">)</mo><mo>∈</mo></mrow><annotation encoding="application/x-tex">C(X,A) \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a> (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 given as a <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a> then this is just the <a class="existingWikiWord" href="/nlab/show/SSet">SSet</a>-<a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a> which is guaranteed to be a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>).</p> <p>Using this, it is useful to reformulate the above as follows slightly:</p> <p> <div class='num_lemma'> <h6>Lemma</h6> <p>An <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <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> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated precisely when for all <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> <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/%28infinity%2C1%29-categorical+hom-space">hom-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\infty Grpd(X,A)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated.</p> </div> </p> <p>In other category-theoretic terms this says that <a class="existingWikiWord" href="/nlab/show/%28n%2Ck%29-transformation">(∞,k)-transformation</a> between <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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, whose components are <a class="existingWikiWord" href="/nlab/show/k-morphisms">k-morphisms</a> in <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>, cannot be nontrivial for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \gt n</annotation></semantics></math> if there are no nontrivial <a class="existingWikiWord" href="/nlab/show/k-morphisms">k-morphisms</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \gt n</annotation></semantics></math> in <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>.</p> <p>Using this fact we can transport the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncation to any <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> by testing it on <a class="existingWikiWord" href="/nlab/show/hom-infinity-groupoid">hom-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-groupoids</a>:</p> <div class="num_defn" id="nTruncatedObject"> <h6 id="definition_3">Definition</h6> <p><strong>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated object in 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>-category)</strong></p> <p>Given an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</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">\mathcal{C}</annotation></semantics></math>, an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math> is called <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated</strong>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, if for all objects <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> the <a class="existingWikiWord" href="/nlab/show/hom-%E2%88%9E-groupoid">hom-∞-groupoid</a> <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>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(X,A)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated according to Def. <a class="maruku-ref" href="#nTruncatedInfinityGroupoid"></a>.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, def. 5.5.6.1</a>.</p> <p>Some terminology:</p> <ul> <li> <p>A 0-truncated object is also called <strong>discrete</strong>. Notice that this is <em>categorically discrete</em> as in <a class="existingWikiWord" href="/nlab/show/discrete+category">discrete category</a>, not discrete in the sense of <a class="existingWikiWord" href="/nlab/show/discrete+topological+space">discrete topological space</a>. An object in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is discrete in this sense if, regarded as an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> with extra structure, it has only trivial morphisms.</p> </li> <li> <p>By the above convention on (-2)-truncated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids, it is only the <a class="existingWikiWord" href="/nlab/show/terminal+object"> terminal objects</a> of <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> that are (-2)-truncated.</p> </li> <li> <p>Similarly, the (-1)-truncated objects are the <a class="existingWikiWord" href="/nlab/show/subterminal+objects">subterminal objects</a>.</p> </li> </ul> <div class="num_defn" id="nTruncatedMorphism"> <h6 id="definition_4">Definition</h6> <p><strong>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated morphism in 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>-category)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated if all of its <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated by def. <a class="maruku-ref" href="#nTruncatedInfinityGroupoid"></a>.</p> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</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> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">W \in C</annotation></semantics></math> the postcomposition morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> C(W,f) : C(W,X) \to C(W,Y) </annotation></semantics></math></div> <p>is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, def. 5.5.6.8</a>.</p> <p>By the <a href="http://ncatlab.org/nlab/show/fiber+sequence#OfFuncCats">characterization of homotopy fiber of functor categories</a> this is equivalent to saying that <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> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated when it is so regarded as an object of the <a class="existingWikiWord" href="/nlab/show/over+quasi-category">slice (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>Y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{/Y}</annotation></semantics></math>. (See also <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, rem. 5.5.6.12</a>.)</p> <p>Unwinding the definitions and applying the <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a>, we have:</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq -1</annotation></semantics></math>, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> of ∞-groupoids is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated iff, for every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_{n+1}(X, x) \to \pi_{n+1}(Y, f(x))</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> and for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mi>n</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">k \geq n+2</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_k(X, x) \to \pi_k(Y, f(x))</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p><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> is <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><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-2)</annotation></semantics></math>-trunacated iff it is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>.</p> </div> <h3 id="in_terms_of_categorical_homotopy_groups">In terms of categorical homotopy groups</h3> <p>At least if the ambient <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> is even an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> there is an alternative, more intrinsic, characterization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncation in terms of <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> <div class="num_prop" id="RecognizngnTuncationOnSimplicialHomotopyGroups"> <h6 id="proposition_2">Proposition</h6> <p>Suppose that an object <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> in an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated for <em>some</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math> (possibly very large).</p> <p>Then for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> this <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated precisely if all the <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups</a> above degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> are trivial.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop 6.5.1.7</a>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Notice that this expected statement does require the assumption that <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. Without any <em>a priori</em> truncation assumption on <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>, there is no comparable statement about the relation to categorical homotopy groups. See <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, remark 6.5.1.8</a>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="RecursiveDefinition">Recursive definition</h3> <div class="num_prop" id="DiagonalCharacterization"> <h6 id="proposition_3">Proposition</h6> <p>In 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>-category <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> with <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \geq -1</annotation></semantics></math>) precisely if the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X \to X \times_Y X</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k-1)</annotation></semantics></math>-truncated.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, lemma 5.5.6.15</a>.</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By definition <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> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated if for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">d \in C</annotation></semantics></math> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(d,f)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>. Since the <a class="existingWikiWord" href="/nlab/show/hom-functor"> hom-functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(d,-)</annotation></semantics></math> preserve <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limits">(∞,1)-limits</a>, we have in particular that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X \to X \times_Y X</annotation></semantics></math> 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> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(d,X) \to C(d,X) \times_{C(d,Y)} C(d,X)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>. Therefore it is sufficient to prove the statement for morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</p> <p>So let now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> be a morphism of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid"> ∞-groupoids</a>. We may find a fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ϕ</mi><mo stretchy="false">¯</mo></mover><mo>:</mo><mover><mi>X</mi><mo stretchy="false">¯</mo></mover><mo>→</mo><mover><mi>Y</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar \phi : \bar X \to \bar Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complexes</a> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> that models <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> in the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>, and by the standard rules for <a class="existingWikiWord" href="/nlab/show/homotopy+pullback"> homotopy pullbacks</a> it follows that the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times_Y X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Grpd is then modeled by the ordinary <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">¯</mo></mover><msub><mo>×</mo> <mover><mi>Y</mi><mo stretchy="false">¯</mo></mover></msub><mover><mi>X</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar X \times_{\bar Y} \bar X</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>. And the homotopy fibers of <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> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">y \in Y</annotation></semantics></math> are then given by the ordinary fibers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>X</mi><mo stretchy="false">¯</mo></mover> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\bar X_y</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar f</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math>.</p> <p>This way, the statement is reduced to the following fact: a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>X</mi><mo stretchy="false">¯</mo></mover> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\bar X_y</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated precisely if the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber"> homotopy fibers</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>X</mi><mo stretchy="false">¯</mo></mover> <mi>y</mi></msub><mo>→</mo><msub><mover><mi>X</mi><mo stretchy="false">¯</mo></mover> <mi>y</mi></msub><mo>×</mo><msub><mover><mi>X</mi><mo stretchy="false">¯</mo></mover> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\bar X_y \to \bar X_y \times \bar X_y</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k-1)</annotation></semantics></math>-truncated.</p> <p>We now write <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> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>X</mi><mo stretchy="false">¯</mo></mover> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\bar X_y</annotation></semantics></math> for simplicity. To see the last statement, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>:</mo><mo>*</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(a,b) : * \to X \times X</annotation></semantics></math> and compute the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></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><mi>Q</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Q &\to& * \\ \downarrow && \downarrow^{\mathrlap{(a,b)}} \\ X &\to& X \times X } </annotation></semantics></math></div> <p>as usual by replacing the right vertical morphism with the fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>I</mi></msup><msub><mo>×</mo> <mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(X \times X)^I \times_{X \times X} (a,b) \to X \times X</annotation></semantics></math> and then forming the ordinary pullback. This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> is equivalent to the space of paths <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">P_{a,b}X</annotation></semantics></math> 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> from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>. (Use that gluing of <a class="existingWikiWord" href="/nlab/show/path+space+object"> path space objects</a> at endpoints of paths produces a new path space; see, for instance, section 4 of <a class="existingWikiWord" href="/nlab/show/BrownAHT">BrownAHT</a>).</p> <p>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 <a class="existingWikiWord" href="/nlab/show/connected">connected</a>, then choosing any path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \to b</annotation></semantics></math> gives an isomorphism from the homotopy groups of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">P_{a,b} X</annotation></semantics></math> to those of the <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>a</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega_a X</annotation></semantics></math>. These latter are indeed those 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>, shifted down in degree by one (as described, for instance, at <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>).</p> <p>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 not connected, we can easily reduce to the case that it is.</p> </div> <h3 id="PropertiesGeneral">General</h3> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>For <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> 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>-category and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">k \geq -2</annotation></semantics></math>, the full <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-category">sub-(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">\tau_{\leq k} C</annotation></semantics></math> is stable under all <a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-limits</a> 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>.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 5.5.6.5</a>.</p> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>. For all <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><mn>2</mn><mo stretchy="false">)</mo><mo>≤</mo><mi>n</mi><mo><</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">(-2) \leq n \lt \infty</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/class">class</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> forms the right class in a <a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+system+in+an+%28%E2%88%9E%2C1%29-category">orthogonal factorization system in an (∞,1)-category</a>. The left class is that of <a class="existingWikiWord" href="/nlab/show/n-connected">n-connected</a> morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> </div> <p>This appears as a remark in <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, Example 5.2.8.16</a>. A construction of the factorization in terms of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentation</a> is in (<a href="#Rezk">Rezk, prop. 8.5</a>). See also <a class="existingWikiWord" href="/nlab/show/n-connected%2Fn-truncated+factorization+system">n-connected/n-truncated factorization system</a>.</p> <h3 id="ModelCategoryPresentations">Model category presentations</h3> <p>There are <a class="existingWikiWord" href="/nlab/show/model+structures+for+homotopy+n-types">model structures for homotopy n-types</a> that <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a> present the full <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-categories">sub-(∞,1)-categories</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated objects in some ambient <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. See there for more details.</p> <h2 id="truncation">Truncation</h2> <h3 id="definition_5">Definition</h3> <p>Under mild conditions there is for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> a universal way to send an arbitrary object <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> to its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\tau_{\leq n} A</annotation></semantics></math>. This is a general version of <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a> where <a class="existingWikiWord" href="/nlab/show/k-morphism">n-morphisms</a> are identified if they are connected by an invertible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-morphism.</p> <p>For <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> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq -2</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">\tau_{\leq n} C </annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <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> on its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated objects.</p> <p>So for instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mn>∞</mn><mi>Grpd</mi><mo>=</mo><mi>n</mi><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\tau_{\leq n} \infty Grpd = n Grpd</annotation></semantics></math>.</p> <div class="num_prop" id="nTruncationReflection"> <h6 id="proposition_6">Proposition</h6> <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 an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> that is <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable</a> then the canonical inclusion <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>C</mi><mo>↪</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> \tau_{\leq n} C \hookrightarrow C </annotation></semantics></math></div> <p>has an <a class="existingWikiWord" href="/nlab/show/accessible+%28infinity%2C1%29-functor">accessible</a> <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mo>:</mo><mi>C</mi><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tau_{\leq n} : C \to \tau_{\leq n} C \,. </annotation></semantics></math></div></div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT 5.5.6.18</a>.</p> <p>Indeed, as the notation suggests, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">\tau_{\leq n} C</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/essential+image">essential image</a> of <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> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\tau_{\leq n}</annotation></semantics></math>. The image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\tau_{\leq n} A</annotation></semantics></math> of an object <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> under this operation is the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncation</strong> of <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>.</p> <p>So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated objects form a <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>C</mi><mover><mo>↪</mo><mover><mo>←</mo><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow></mover></mover><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tau_{\leq n} C \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow} C \,. </annotation></semantics></math></div> <h3 id="Properties">Properties</h3> <h4 id="GeneralPropsTruncation">General</h4> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p>For any small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>n</mi><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{\leq n} PSh(C, \infty Grpd) \simeq PSh(C, n Grpd) </annotation></semantics></math>, and truncation acts pointwise.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>If <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> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated ∞-presheaf, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(c) \simeq PSh(C)(C(-, c), P)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated; thus <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> takes values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">n Grpd</annotation></semantics></math>.</p> <p>Conversely, if <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> takes values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">n Grpd</annotation></semantics></math>, then the fact every presheaf is a colimit of representables implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>Q</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom(Q, P)</annotation></semantics></math> is a limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated spaces and is thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated.</p> <p>Given this identification of the subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated objects, we can see that the truncation-inclusion adjunction between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">n Grpd</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> induces an adjunction whose right adjoint is the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>n</mi><mi>Grpd</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(C, n Grpd) \to PSh(C, \infty Grpd)</annotation></semantics></math></p> </div> <p> <div class='num_cor'> <h6>Corollary</h6> <p><strong>(left exact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-functors preserve truncation)</strong> <br /> A <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a> <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-functor"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F \colon C \to D</annotation></semantics></math> between <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>-categories with <a class="existingWikiWord" href="/nlab/show/finite+%28infinity%2C1%29-limit">finite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-limits</a> sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated objects/morphisms to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated objects/morphisms.</p> </div> (<a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 5.5.6.16</a>) <div class='proof'> <h6>Proof</h6> <p>This follows from the above recursive characterization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-truncated morphisms by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k-1)</annotation></semantics></math>-truncation of their diagonal, which is preserved by the finite limit preserving <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>.</p> </div> </p> <div class="num_prop"> <h6 id="proposition_8">Proposition</h6> <p>A left exact presentable <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>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F : C \to D</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-categories">locally presentable (∞,1)-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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> commutes with truncation:</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><msubsup><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow> <mi>C</mi></msubsup><mo>≃</mo><msubsup><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow> <mi>D</mi></msubsup><mo>∘</mo><mi>F</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F \circ \tau^C_{\leq k} \simeq \tau^D_{\leq k} \circ F \,. </annotation></semantics></math></div></div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 5.5.6.28</a>.</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By the above lemma, <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> restricts to a functor on the truncations. So we need to show that the diagram</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><mi>C</mi></mtd> <mtd><mover><mo>→</mo><mi>F</mi></mover></mtd> <mtd><mi>D</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mo>?</mo><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub><mi>C</mi></mtd> <mtd><mover><mo>→</mo><mi>F</mi></mover></mtd> <mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C &\stackrel{F}{\to}& D \\ {}^{\mathllap{\tau_{\leq k}}}\downarrow & (?) & \downarrow^{\mathrlap{\tau_{\leq k}}} \\ \tau_{\leq k } C &\stackrel{F}{\to}& \tau_{\leq k} D } </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a> can be filled by a 2-cell. To see this, notice that the <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a> of both composite morphisms exists (because that of <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> exists by the <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a> and bcause adjoints of composites are composites of adjoints) and since the bottom morphism is just the restriction of the top morphism and the right adjoints of the vertical morphisms are full inclusions this adjoint diagram</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><mi>C</mi></mtd> <mtd><mover><mo>←</mo><mi>G</mi></mover></mtd> <mtd><mi>D</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub><mi>C</mi></mtd> <mtd><mover><mo>←</mo><mi>G</mi></mover></mtd> <mtd><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub><mi>D</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ C &\stackrel{G}{\leftarrow}& D \\ \uparrow & & \uparrow \\ \tau_{\leq k } C &\stackrel{G}{\leftarrow}& \tau_{\leq k} D } \,. </annotation></semantics></math></div> <p>evidently commutes since it just expresses this restriction.</p> </div> <div class="num_prop" id="nTruncationInToposPreservesFiniteProducts"> <h6 id="proposition_9">Proposition</h6> <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 an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, then truncation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\tau_{\leq n} : C \to C</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limit">finite</a> <a class="existingWikiWord" href="/nlab/show/products">products</a>.</p> </div> <p>This appears as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, lemma 6.5.1.2</a>.</p> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>First notice that the statement is true for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>. For instance we can use the example <a href="#InInfGrpd">In ∞Grpd and Top</a>, model <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoids</a> by <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complexes</a> and notice that then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\tau_{\leq n}</annotation></semantics></math> is given by the truncation functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>tr</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>:</mo><mi>sSet</mi><mo>→</mo><mo stretchy="false">[</mo><msubsup><mi>Δ</mi> <mrow><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mi>op</mi></msubsup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">tr_{n+1} : sSet \to [\Delta^{op}_{\leq n+1}, Set]</annotation></semantics></math>. This is also a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> and as such preserves in particular products in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math>, which are <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>-products in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math>.</p> <p>From that we deduce that the statement is true for <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> any <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Func</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msup><mi>K</mi> <mi>op</mi></msup><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = PSh_{(\infty,1)}(K) = Func_{(\infty,1)}(K^{op}, \infty Grpd)</annotation></semantics></math> because all relevant operations there are objectwise those in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math>.</p> <p>So far, this shows even that on presheaf <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>-toposes, all products (not necessarily finite) are preserved by truncation.</p> <p>A general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</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> is (by definition) a left exact <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a> of a presheaf <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>-topos,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mover><munder><mo>↪</mo><mi>i</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C \stackrel{\overset{L}{\leftarrow}}{\underset{i}{\hookrightarrow}} PSh_{(\infty,1)}(K) \,. </annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>j</mi></msub><mi>i</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{j} i(X_j)</annotation></semantics></math> be the product of the objects in question taken in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(K)</annotation></semantics></math>. By the above there, we have an equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>j</mi></munder><mi>i</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>j</mi></munder><msub><mi>τ</mi> <mo>≤</mo></msub><mi>i</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tau_{\leq k} \prod_j i(X_j) \stackrel{\simeq}{\to} \prod_j \tau_{\leq} i(X_j) \,. </annotation></semantics></math></div> <p>Now applying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> to this equivalence and using now that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves the <em>finite</em> product, this gives an equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>j</mi></munder><mi>i</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>L</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>j</mi></munder><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub><mi>i</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>j</mi></munder><mi>L</mi><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msub><mi>i</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L \tau_{\leq k} \prod_j i(X_j) \stackrel{\simeq}{\to} L \prod_j \tau_{\leq k} i(X_j) \simeq \prod_j L \tau_{\leq k} i(X_j) </annotation></semantics></math></div> <p>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>. The claim follows now with the above result that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∘</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mo>≃</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mo>∘</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">L \circ \tau_{\leq n} \simeq \tau_{\leq n} \circ L</annotation></semantics></math>.</p> </div> <h4 id="postnikov_tower">Postnikov tower</h4> <p>By the fact that the truncation functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\tau_{\leq n}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>, one obtains canonical morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>A</mi></mrow><annotation encoding="application/x-tex"> A \to \tau_{\leq n}A </annotation></semantics></math></div> <p>as the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> on <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 then by iteration also canonical morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mi>A</mi><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tau_{\leq (n+1)} A \to \tau_{\leq n} A \,. </annotation></semantics></math></div> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \in C</annotation></semantics></math>, the sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>2</mn></mrow></msub><mi>A</mi><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>1</mn></mrow></msub><mi>A</mi><mo>→</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mn>0</mn></mrow></msub><mi>A</mi></mrow><annotation encoding="application/x-tex"> \cdots \to \tau_{\leq 2}A \to \tau_{\leq 1} A \to \tau_{\leq 0} A </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower in an (∞,1)-category</a> of <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>. See there for more details.</p> <h4 id="TruncationHomotopyTypeTheorySyntax">Homotopy type theory syntax</h4> <p>Discussion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncation of <a class="existingWikiWord" href="/nlab/show/types">types</a> in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> via <a class="existingWikiWord" href="/nlab/show/higher+inductive+types">higher inductive types</a> is in (<a href="#Brunerie">Brunerie</a>). This sends a type to an <a class="existingWikiWord" href="/nlab/show/h-level">h-level</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+2)</annotation></semantics></math>-type. The <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><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1)</annotation></semantics></math>-truncation in the context is forming the <a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> <a class="existingWikiWord" href="/nlab/show/hProp">hProp</a>.</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></em>.</p> <h3 id="RelationToHomotopyGroups">Relation to homotopy groups</h3> <p>In 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>-topos <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> there is a notion of categorical <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy groups in an (∞,1)-topos</a>. For the <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>-topos <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> given by the model of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complexes</a> this coincides with the notion of <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+groups">simplicial homotopy groups</a>:</p> <div class="num_lemma"> <h6 id="observation">Observation</h6> <p>An object <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> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated precisely if its <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_k(A)</annotation></semantics></math> vanish for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \gt n</annotation></semantics></math>.</p> </div> <p>This simple relation between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncation and categorical homotopy groups is almost, but not exactly, true in an arbitrary <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>.</p> <div class="num_prop"> <h6 id="proposition_10">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">A \in \mathbf{H}</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated object.</p> <p>Then</p> <ol> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \gt n</annotation></semantics></math> we have for the <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\pi_k(A) = *</annotation></semantics></math>;</p> </li> <li> <p>if (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n \geq 0</annotation></semantics></math>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\pi_n(A) = *</annotation></semantics></math>, then <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 in fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-truncated.</p> </li> </ol> </div> <p>This implies</p> <div class="num_corollary"> <h6 id="corollary">Corollary</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">A \in \mathbf{H}</annotation></semantics></math> is truncated at all (for any value), then it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated precisely if all categorical homotopy groups vanish <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\pi_k(A) = *</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \gt n</annotation></semantics></math>.</p> </div> <p><strong>Notice.</strong> If <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>, on the other hand, is not truncated at all, then all its homotopy groups may be trivial, and <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> may still not be equivalent to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>. This means that <a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a> may fail in a general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> for untruncated objects. It holds, however, in <a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-toposes</a>.</p> <h2 id="Examples">Examples</h2> <h3 id="truncated_morphisms">Truncated morphisms</h3> <h4 id="general_3">General</h4> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f : X \to 0</annotation></semantics></math> is</p> <ul> <li> <p>(-2)-truncated precisely if it is an <a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalence</a>;</p> </li> <li> <p>(-1)-truncated precisely if it is a <a class="existingWikiWord" href="/nlab/show/monomorphism+in+an+%28infinity%2C1%29-category">monomorphism</a>.</p> </li> </ul> <h4 id="TruncatedMorphismsBetweenGroupoids">Between groupoids</h4> <p>For morphisms between 1-<a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>, the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncation for low <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> reproduces standard concepts from ordinary <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>.</p> <div class="num_prop" id="TruncatedFunctorsOfGroupoids"> <h6 id="proposition_11">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>, is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated precisely when regarded as a morphism in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> it is</p> <ul> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = -2</annotation></semantics></math> – an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>;</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = -1 </annotation></semantics></math> – a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a>;</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> – a <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful functor</a>.</p> </li> </ul> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>Notice that <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> being <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful</a> means precisely that it induces a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> on the first <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x : * \to X</annotation></semantics></math> any point and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">F_{f(x)}</annotation></semantics></math> the corresponding <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <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>, the <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a> gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(F)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of an <a class="existingWikiWord" href="/nlab/show/injective">injective</a> map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \cdots \to \pi_1(F) \to \pi_1(X) \hookrightarrow \pi_1(Y,f(x)) \to \cdots \,, </annotation></semantics></math></div> <p>hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>y</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\pi_1(F_{y}) = *</annotation></semantics></math> for all points <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> in the essential image of <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>. For <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> not in the essential image we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>y</mi></msub><mo>≃</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">F_y \simeq \emptyset</annotation></semantics></math>. In either case, it follows that <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> is 0-truncated.</p> <p>By def. <a class="maruku-ref" href="#nTruncatedMorphism"></a> this is the defining condition for <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> to be 0-truncated.</p> </div> <h4 id="TruncatedMorphismsBetweenStacks">Between stacks</h4> <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">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(2,1)}(C) \hookrightarrow Sh_{(\infty,1)}(C)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>/<a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaves">(2,1)-sheaves</a> inside the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-sheaf (∞,1)-topos</a> of all <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a>/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>W</mi></msub><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Grpd</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">L_W [C^{op}, Grpd]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> of groupoid valued presheaves 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> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj,loc}</annotation></semantics></math> for the local projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> that <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presents</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(\infty,1)}(C)</annotation></semantics></math>.</p> <div class="num_prop"> <h6 id="proposition_12">Proposition</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>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> be a morphism of stacks that has a presentation by a degreewise <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful functor</a> that, under the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>, goes between fibrant <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a>.</p> <p>Then <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> is 0-truncated as a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(\infty,1)}(C)</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>We need to check that for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack <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>, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_\infty(A,f)</annotation></semantics></math> is 0-truncated in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>. We may choose a cofibrant model for <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[C^{op}, sSet]_{proj,loc}</annotation></semantics></math> and by assumption that <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> is fibrant we have that the ordinary hom of simplicial presheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[C^{op}, sSet](A, f) </annotation></semantics></math> is the correct <a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a> morphism. This is itself (the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of) a <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful functor</a>, hence the statement follows with prop. <a class="maruku-ref" href="#TruncatedFunctorsOfGroupoids"></a>.</p> </div> <h3 id="InInfGrpd">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> and 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></h3> <p>An object in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated precisely if it is an <a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a>. To some extent, this is so by definition. Equivalently, an object in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated if it is (in the equivalence class of) a <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a>.</p> <p>So we have for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>n</mi><mi>Grpd</mi><mover><mo>↪</mo><mover><mo>←</mo><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow></mover></mover><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> n Grpd \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow} \infty Grpd \,. </annotation></semantics></math></div> <div class="num_lemma"> <h6 id="observation_2">Observation</h6> <p>If we model the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> as the <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>/fibrant <a class="existingWikiWord" href="/nlab/show/simplicial+category">simplicial category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KanCplx</mi><mo>⊂</mo></mrow><annotation encoding="application/x-tex">KanCplx \subset </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complexes</a>, then the truncation adjunction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub><mo>⊣</mo><mi>i</mi><mo stretchy="false">)</mo><mo>:</mo><mi>n</mi><mi>Grpd</mi><mover><mo>↪</mo><mover><mo>←</mo><mrow><msub><mi>τ</mi> <mrow><mo>≤</mo><mi>n</mi></mrow></msub></mrow></mover></mover><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\tau_{\leq n } \dashv i) : n Grpd \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow} \infty Grpd \,. </annotation></semantics></math></div> <p>is modeled by the <a class="existingWikiWord" href="/nlab/show/simplicial+skeleton">simplicial coskeleton</a> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-enriched adjunction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>tr</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⊣</mo><msub><mi>cosk</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mi>KanCplx</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mover><munder><mo>→</mo><mrow><msub><mi>cosk</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></munder><mover><mo>←</mo><mrow><msub><mi>tr</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mover></mover><mi>KanCplx</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (tr_{n+1} \dashv cosk_{n+1}) : KanCplx_{n+1} \stackrel{\overset{tr_{n+1}}{\leftarrow}}{\underset{cosk_{n+1}}{\to}} KanCplx \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>KanCplx</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">KanCplx_{n+1}</annotation></semantics></math> is the subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>Δ</mi> <mrow><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mi>op</mi></msubsup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta^{op}_{\leq n+1}, Set]</annotation></semantics></math> on those truncated simplicial sets that are truncations of Kan complexes, regarded as a Kan-complex-enriched category by the embedding via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cosk</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">cosk_{n+1}</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>Notice that every <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <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> which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated is homotopy equivalent to one in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cosk</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">cosk_{n+1}</annotation></semantics></math>, namely to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cosk</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>tr</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">cosk_{n+1} tr_{n+1} X</annotation></semantics></math>, because by one of the <a href="http://ncatlab.org/nlab/show/simplicial+skeleton#Truncation">properties</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cosk</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">cosk_{n+1}</annotation></semantics></math> we have that the unit</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>cosk</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>tr</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \to cosk_{n+1} tr_{n+1} X </annotation></semantics></math></div> <p>induces isomorphisms on homotopy groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\pi_k</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \leq n</annotation></semantics></math>.</p> <p>This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>KanCplx</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">KanCplx_{n+1}</annotation></semantics></math> is indeed a full <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-category">sub-(∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KanCplx</mi></mrow><annotation encoding="application/x-tex">KanCplx</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated objects</p> <p>Moreover, by the fact discussed at <a href="http://ncatlab.org/nlab/show/adjoint+(infinity%2C1)-functor#SimplicialAndDerived">Simplicial and derived adjunctions</a> at <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functor</a> we have that the <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-enriched adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>tr</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⊣</mo><msub><mi>cosk</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(tr_{n+1} \dashv cosk_{n+1})</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KanCplx</mi></mrow><annotation encoding="application/x-tex">KanCplx</annotation></semantics></math> indeed presents a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint (∞,1)-functors</a> on <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>. So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>tr</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>:</mo><mi>KanCplx</mi><mo>→</mo><mi>KanCplx</mi></mrow><annotation encoding="application/x-tex">tr_{n+1} : KanCplx \to KanCplx</annotation></semantics></math> indeed presents the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mo>≤</mo></msub><mo>:</mo><mn>∞</mn><mi>Grpd</mi><mo>→</mo><mi>n</mi><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\tau_{\leq} : \infty Grpd \to n Grpd</annotation></semantics></math> to the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mi>Grpd</mi><mo>↪</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">n Grpd \hookrightarrow \infty Grpd</annotation></semantics></math>.</p> </div> <h3 id="DiagonalExamples">Diagonals</h3> <div class="num_example"> <h6 id="example">Example</h6> <p>In ordinary <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> we have that a morphism is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> (as discussed there), precisely if its <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. Embedded into <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>, this becomes the special case of prop. <a class="maruku-ref" href="#DiagonalCharacterization"></a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math>: a morphism is (-1)-truncated (hence a <a class="existingWikiWord" href="/nlab/show/monomorphism+in+an+%28%E2%88%9E%2C1%29-category">monomorphism in an (∞,1)-category</a>), precisely if its diagonal is (-2)-truncated (hence an <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence in an (∞,1)-category</a>).</p> </div> <div class="num_example"> <h6 id="example_2">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 an object that is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to *</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated morphism. So by prop. <a class="maruku-ref" href="#DiagonalCharacterization"></a> the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> on that object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \Delta : X \to X \times X </annotation></semantics></math></div> <p>is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-truncated morphism, and precisely if it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-truncated is <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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated.</p> <p>In particular, the diagonal is a <a class="existingWikiWord" href="/nlab/show/monomorphism+in+an+%28%E2%88%9E%2C1%29-category">monomorphism in an (∞,1)-category</a>, hence (-1)-truncated, precisely 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-truncated (an <a class="existingWikiWord" href="/nlab/show/h-set">h-set</a>).</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></p> </li> <li> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated morphism</strong> / <a class="existingWikiWord" href="/nlab/show/n-connected">n-connected</a> morphism</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+homotopy+n-types">model structure for homotopy n-types</a></p> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/homotopy+level">homotopy level</a></th><th><a class="existingWikiWord" href="/nlab/show/n-truncated+object+in+an+%28infinity%2C1%29-category">n-truncation</a></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos+theory">higher topos theory</a></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></th></tr></thead><tbody><tr><td style="text-align: left;">h-level 0</td><td style="text-align: left;">(-2)-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28-2%29-groupoid">(-2)-groupoid</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/true">true</a>/<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a>/<a class="existingWikiWord" href="/nlab/show/contractible+type">contractible type</a></td></tr> <tr><td style="text-align: left;">h-level 1</td><td style="text-align: left;">(-1)-truncated</td><td style="text-align: left;">contractible-if-<a class="existingWikiWord" href="/nlab/show/inhabited+space">inhabited</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28-1%29-groupoid">(-1)-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-sheaf">(0,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/ideal">ideal</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a>/<a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a></td></tr> <tr><td style="text-align: left;">h-level 2</td><td style="text-align: left;">0-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+0-type">homotopy 0-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/0-groupoid">0-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-set">h-set</a></td></tr> <tr><td style="text-align: left;">h-level 3</td><td style="text-align: left;">1-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+1-type">homotopy 1-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/1-groupoid">1-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/stack">stack</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-groupoid">h-groupoid</a></td></tr> <tr><td style="text-align: left;">h-level 4</td><td style="text-align: left;">2-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+2-type">homotopy 2-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></td><td style="text-align: left;">(3,1)-sheaf/2-stack</td><td style="text-align: left;">h-2-groupoid</td></tr> <tr><td style="text-align: left;">h-level 5</td><td style="text-align: left;">3-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+3-type">homotopy 3-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></td><td style="text-align: left;">(4,1)-sheaf/3-stack</td><td style="text-align: left;">h-3-groupoid</td></tr> <tr><td style="text-align: left;">h-level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n+2</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><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28n%2B1%2C1%29-sheaf">(n+1,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/n-stack">n-stack</a></td><td style="text-align: left;">h-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoid</td></tr> <tr><td style="text-align: left;">h-level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></td><td style="text-align: left;">untruncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a></td><td style="text-align: left;">h-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid</td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="in_category_theory">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category theory</h3> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+theory"><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</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, Section 5.5.6 of: <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> <p>Discussion of <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> is in section 6.5.1 there.</p> <p>Discussion in terms of <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentations</a>:</p> <ul> <li id="Rezk"><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, Section 7 of: <em>Toposes and homotopy toposes</em>, 2010 (<a href="http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Rezk_HomotopyToposes.pdf" title="pdf">pdf</a>)</li> </ul> <p>A classical article that amplifies the expression of <a class="existingWikiWord" href="/nlab/show/Postnikov+towers">Postnikov towers</a> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> (hence in <a class="existingWikiWord" href="/nlab/show/Infinity-Grpd"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>Grpd</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">Grpd_{\infty}</annotation> </semantics> </math></a>) in terms of <a class="existingWikiWord" href="/nlab/show/coskeleta">coskeleta</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Kan">Dan Kan</a>, <em>An obstruction theory for diagrams of simplicial sets</em> (<a href="http://www.nd.edu/~wgd/Dvi/ObstructionTheoryForDiagrams.pdf">pdf</a>)</li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal</a> <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, section 7.6 of <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em></p> </li> <li id="Brunerie"> <p><a class="existingWikiWord" href="/nlab/show/Guillaume+Brunerie">Guillaume Brunerie</a>, <em>Truncations and truncated higher inductive types</em> (<a href="http://homotopytypetheory.org/2012/09/16/truncations-and-truncated-higher-inductive-types/">web</a>)</p> </li> </ul> <p>More discussion of the internal perspective:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nima+Rasekh">Nima Rasekh</a>, <em>An Elementary Approach to Truncations</em> (<a href="https://arxiv.org/abs/1812.10527">arXiv:1812.10527</a>)</li> </ul> <h3 id="in_homotopy_type_theory">In homotopy type theory</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/homotopy+n-types">homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-types</a>/<a class="existingWikiWord" href="/nlab/show/n-truncated+object+in+an+%28infinity%2C1%29-category"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-truncated objects</a> in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, §7 of: <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em> (2013) [<a href="http://homotopytypetheory.org/book/">web</a>, <a href="http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicolai+Kraus">Nicolai Kraus</a>, <em>Truncation levels in Homotopy Type Theory</em>, Nottingham (2015) [<a href="https://eprints.nottingham.ac.uk/28986/1/thesis.pdf">pdf</a>, <a href="https://eprints.nottingham.ac.uk/28986">eprints:28986</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Felix+Cherubini">Felix Cherubini</a>, <a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, Thm. 3.10 of: <em>Modal Descent</em>, Mathematical Structures in Computer Science , <strong>31</strong> 4 (2021) 363-391 [<a href="https://doi.org/10.1017/S0960129520000201">doi:10.1017/S0960129520000201</a>, <a href="https://arxiv.org/abs/2003.09713">arXiv:2003.09713</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 21, 2025 at 14:00:29. See the <a href="/nlab/history/n-truncated+object+of+an+%28infinity%2C1%29-category" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/n-truncated+object+of+an+%28infinity%2C1%29-category" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/11353/#Item_8">Discuss</a><span class="backintime"><a href="/nlab/revision/n-truncated+object+of+an+%28infinity%2C1%29-category/81" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/n-truncated+object+of+an+%28infinity%2C1%29-category" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/n-truncated+object+of+an+%28infinity%2C1%29-category" accesskey="S" class="navlink" id="history" rel="nofollow">History (81 revisions)</a> <a href="/nlab/show/n-truncated+object+of+an+%28infinity%2C1%29-category/cite" style="color: black">Cite</a> <a href="/nlab/print/n-truncated+object+of+an+%28infinity%2C1%29-category" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/n-truncated+object+of+an+%28infinity%2C1%29-category" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>