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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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 theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" 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> </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='#ViaEpiMonoFactorization'>By epi/mono factorization 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</a></li> <li><a href='#ViaColimitOfCechNerve'>As the ∞-colimit of the kernel ∞-groupoid</a></li> <li><a href='#tower_of_images'>Tower 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>-images</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#SyntaxInHomotopyTypeTheory'>Syntax in homotopy type theory</a></li> <li><a href='#compatibility_with_limits'>Compatibility with limits</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#automorphisms'>Automorphisms</a></li> <li><a href='#NImagesOf1FunctorsBetweenGroupoids'>Of functors between groupoids</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The generalization of the notion of <em><a class="existingWikiWord" href="/nlab/show/image">image</a></em> from <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>.</p> <h2 id="definition">Definition</h2> <h3 id="ViaEpiMonoFactorization">By epi/mono factorization 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</h3> <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>. Then by the discussion at <em><a class="existingWikiWord" href="/nlab/show/%28n-connected%2C+n-truncated%29+factorization+system">(n-connected, n-truncated) factorization system</a></em> there is a <a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+system+in+an+%28%E2%88%9E%2C1%29-category">orthogonal factorization system</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Epi</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>,</mo><mi>Mono</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>⊂</mo><mspace width="thinmathspace"></mspace><mi>Mor</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>×</mo><mi>Mor</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \big( Epi(\mathbf{H}), Mono(\mathbf{H}) \big) \,\subset\, Mor(\mathbf{H}) \times Mor(\mathbf{H}) </annotation></semantics></math></div> <p>whose left class consists of the <a class="existingWikiWord" href="/nlab/show/n-connected+object+in+an+%28%E2%88%9E%2C1%29-topos">(-1)-connected</a> morphisms, the <a class="existingWikiWord" href="/nlab/show/effective+epimorphism+in+an+%28%E2%88%9E%2C1%29-category">(∞,1)-effective epimorphisms</a>, while the right class consists of the <a class="existingWikiWord" href="/nlab/show/n-truncated+object+in+an+%28%E2%88%9E%2C1%29-category">(-1)-truncated morphisms</a> morphisms, hence the <a class="existingWikiWord" href="/nlab/show/monomorphism+in+an+%28%E2%88%9E%2C1%29-category">(∞,1)-monomorphisms</a>.</p> <p>So given 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 <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> with epi-mono factorization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> f : X \to im_1(f) \hookrightarrow Y \,, </annotation></semantics></math></div> <p>we may call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">im_1(f) \hookrightarrow Y</annotation></semantics></math> the <strong>image</strong> 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>.</p> <h3 id="ViaColimitOfCechNerve">As the ∞-colimit of the kernel ∞-groupoid</h3> <p>In a sufficiently well-behaved 1-<a class="existingWikiWord" href="/nlab/show/category">category</a>, the <a class="existingWikiWord" href="/nlab/show/coimage">(co)image</a> of a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> may be defined as the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> of its <a class="existingWikiWord" href="/nlab/show/kernel+pair">kernel pair</a>, hence by the fact that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>X</mi><mover><mo>⟶</mo><mrow></mrow></mover><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X \times_Y X \stackrel{\longrightarrow}{\longrightarrow} X \stackrel{}{\longrightarrow} im(f) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/colimit">colimiting</a> <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> under the <a class="existingWikiWord" href="/nlab/show/parallel+morphism">parallel morphism</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>.</p> <p>In an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> the 1-image is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a> not just of these two morphisms, but of the full “<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>-kernel”, namely of the full <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mover><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover><mo>⟶</mo></mover><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi><mover><mover><mo>⟶</mo><mo>⟶</mo></mover><mo>⟶</mo></mover><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>X</mi><mover><mo>⟶</mo><mrow></mrow></mover><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} X \times_Y X \times_Y X \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} X \times_Y X \stackrel{\longrightarrow}{\longrightarrow} X \stackrel{}{\longrightarrow} im(f) \,. </annotation></semantics></math></div> <p>(Here all degeneracy maps are notionally suppressed.)</p> <p>To see that this gives the same notion of image as given by the epi-mono factorization as discussed <a href="#ViaEpiMonoFactorization">above</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \stackrel{f}{\longrightarrow} im(f) \hookrightarrow Y</annotation></semantics></math> be such a factorization. Then using (by the discussion at <em><a href="n-truncated+object+of+an+%28infinity%2C1%29-category#RecursiveDefinition">truncated morphism – Recursive characterization</a></em>) that the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> of a <a class="existingWikiWord" href="/nlab/show/monomorphism+in+an+%28%E2%88%9E%2C1%29-category">monomorphism</a> is its domain, we find a <a class="existingWikiWord" href="/nlab/show/pasting+diagram">pasting diagram</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> squares of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><msub><mo>×</mo> <mrow><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X \times_{im(f)} X &\to & X &\stackrel{\simeq}{\to} & X \\ \downarrow && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f}} \\ X &\stackrel{f}{\to}& im(f) &\stackrel{\simeq}{\to}& im(f) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow \\ X &\stackrel{f}{\to}& im(f) &\hookrightarrow& Y } \,, </annotation></semantics></math></div> <p>which shows, by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>, that <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><mo>≃</mo><mi>X</mi><msub><mo>×</mo> <mrow><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times_{Y} X \simeq X \times_{im(f)} X</annotation></semantics></math> and hence that the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\to} Y</annotation></semantics></math> is equivalently that of the effective epimorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\to} im(f)</annotation></semantics></math>. Now, by one of the <a class="existingWikiWord" href="/nlab/show/Giraud-Rezk-Lurie+axioms">Giraud-Rezk-Lurie axioms</a> satisfied by <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a>, the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a> over the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> of an <a class="existingWikiWord" href="/nlab/show/effective+epimorphism+in+an+%28%E2%88%9E%2C1%29-category">effective epimorphism</a> is that morphism itself. Therefore the 1-image defined this way coincides with the one defined by the epi/mono factorization.</p> <h3 id="tower_of_images">Tower 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>-images</h3> <p>More generally for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> a morphism we have a <a class="existingWikiWord" href="/nlab/show/tower">tower</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>-images</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>im</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mo>→</mo><msub><mi>im</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>im</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> X \simeq im_\infty(f) \to \cdots \to im_2(f) \to im_1(f) \to im_0(f) \simeq Y </annotation></semantics></math></div> <p>factoring <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>, such that for each <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> the factorization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>im</mi> <mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to im_{n+2}(f) \to Y</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%28n-connected%2C+n-truncated%29+factorization+system">(n-connected, n-truncated) factorization system</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>.</p> <p>This is the relative <a class="existingWikiWord" href="/nlab/show/Postnikov+system">Postnikov system</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>.</p> <h2 id="properties">Properties</h2> <h3 id="SyntaxInHomotopyTypeTheory">Syntax in homotopy type theory</h3> <p>Let the ambient <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> be an <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><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. Then its <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a> is <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>. We discuss the <a class="existingWikiWord" href="/nlab/show/syntax">syntax</a> of 1-images in this theory.</p> <div class="num_prop" id="SyntaxOfInfinityImage"> <h6 id="proposition">Proposition</h6> <p>If</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 lspace="verythinmathspace">:</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>B</mi></mrow><annotation encoding="application/x-tex"> a \colon A \;\vdash\; f(a) \,\colon\, B </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/term">term</a> whose <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> is 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>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A \xrightarrow{f} B</annotation></semantics></math> 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>, then the 1-image of that morphism, when regarded as an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\mathbf{H}/B</annotation></semantics></math>, is the <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> of the <a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></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><mi>b</mi><mo>:</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">[</mo><mo stretchy="false">(</mo><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>×</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>b</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><msub><mo maxsize="1.8em" minsize="1.8em">]</mo> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Type</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (b:B) \;\; \vdash \;\; \Big[ (a \colon A) \times \big(b = f(a) \big) \Big]_0 \,\colon\, Type \,, </annotation></semantics></math></div></div> <p>Here</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>C</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">(a \colon A) \times C_a</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/dependent+pair+type">dependent pair type</a>,</p> </li> <li> <p>“<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math>” denotes the <a class="existingWikiWord" href="/nlab/show/identification+type">identification type</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><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">[-]_0</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> (which is constructible in homotopy type theory either as a <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a> as described at <em><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></em>m or using the <a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a> and a resizing rule to obtain a <a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>).</p> </li> </ul> <div class="proof"> <h6 id="proof">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>M</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{M}</annotation></semantics></math> be a suitable <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">presenting</a> <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>. Then by the rules for <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> of <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a> and <a class="existingWikiWord" href="/nlab/show/substitution">substitution</a>, the interpretation of</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><mi>b</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>b</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>:</mo><mspace width="thinmathspace"></mspace><mi>Type</mi></mrow><annotation encoding="application/x-tex"> (b \colon B) ,\, (a \colon A) \;\;\vdash\;\; \big( b = f(a) \big) \,:\, Type </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>M</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{M}</annotation></semantics></math> a is the following <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>A</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde A</annotation></semantics></math> (see <em><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></em> for more details):</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><mover><mi>A</mi><mo stretchy="false">˜</mo></mover></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>B</mi> <mi>I</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi><mo>×</mo><mi>B</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><msub><mi>id</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>B</mi><mo>×</mo><mi>B</mi></mtd></mtr></mtable></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde A &\to& B^{I} \\ \downarrow && \downarrow \\ A \times B &\stackrel{(f,id_B)}{\to}& B \times B }. </annotation></semantics></math></div> <p>Here all objects now denote <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a> representatives of the given objects 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>, and the right-hand morphism is the <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a> out of a <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> for <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>. By the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">˜</mo></mover><mo>→</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\tilde A \to A \times B \to B</annotation></semantics></math> here is a <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a> <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> of the original <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A \stackrel{f}{\to} B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">˜</mo></mover><mo>→</mo><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\tilde A \to A \times B</annotation></semantics></math> is a fibration resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>A</mi></msub><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \stackrel{(id_A,f)}{\to} A \times B</annotation></semantics></math>. Regarded in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>M</mi></mstyle><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{M}/(A \times B)</annotation></semantics></math>, this now interprets the syntax <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(b = f(a))</annotation></semantics></math> as an <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></mrow><annotation encoding="application/x-tex">(A \times B)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a>.</p> <p>Now the interpretation of the sum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\sum_{a:A}</annotation></semantics></math> is simply that we forget the map to <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> (or equivalently compose with the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\times B\to B</annotation></semantics></math>), regarding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde A</annotation></semantics></math> as an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>M</mi></mstyle><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\mathbf{M}/B</annotation></semantics></math>. Of course, this is just a fibration resolution 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> itself.</p> <p>Finally, the interpretation of the <a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of this is precisely the <a class="existingWikiWord" href="/nlab/show/n-truncated+object+in+an+%28infinity%2C1%29-category">(-1)-truncation</a> of this morphism, which by the discussion there is its 1-image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mover><mi>A</mi><mo stretchy="false">˜</mo></mover><mo>→</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im_1(\tilde A \to B)</annotation></semantics></math>, regarded as a dependent type over <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>. Thus, it is precisely the the 1-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>.</p> </div> <p>By additionally forgetting the remaining map 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>, we obtain:</p> <div class="num_cor" id="NonDependentSyntax"> <h6 id="corollary">Corollary</h6> <p>In the above situation, the 1-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>, regarded as an object of <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> itself, is the semantics of the non-dependent type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⊢</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>b</mi><mo>:</mo><mi>B</mi></mrow></munder><mo maxsize="1.8em" minsize="1.8em">[</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></munder><mo stretchy="false">(</mo><mi>b</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">]</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mo>:</mo><mi>Type</mi><mo maxsize="1.8em" minsize="1.8em">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \vdash \; \Big(\Big( \sum_{b:B}\Big[\sum_{a:A} (b = f(a))\Big] \Big) : Type\Big). </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The bracket type of a <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> is the <a class="existingWikiWord" href="/nlab/show/propositions+as+some+types">propositions as some types</a> version of the <a class="existingWikiWord" href="/nlab/show/existential+quantifier">existential quantifier</a>, so we can write the dependent type in Prop. <a class="maruku-ref" href="#SyntaxOfInfinityImage"></a> as</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><mi>b</mi><mo>:</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo>∃</mo><mi>a</mi><mo>:</mo><mi>A</mi><mo>.</mo><mo stretchy="false">(</mo><mi>b</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>:</mo><mspace width="thickmathspace"></mspace><mi>hProp</mi><mo>)</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex"> (b:B) \; \vdash \; \left(\exists a:A . (b = f(a)) \;:\; hProp\right). </annotation></semantics></math></div> <p>The dependent sum <em>of</em> an <a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a> is then the propositions-as-some-types version of the <a class="existingWikiWord" href="/nlab/show/comprehension+rule">comprehension rule</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>b</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">|</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{b \in B | \phi(b)\}</annotation></semantics></math>, so the non-dependent type in Cor. <a class="maruku-ref" href="#NonDependentSyntax"></a> may be written as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mi>b</mi><mo>∈</mo><mi>B</mi><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mo>∃</mo><mi>a</mi><mo>∈</mo><mi>A</mi><mo>.</mo><mo stretchy="false">(</mo><mi>b</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ b \in B \,|\, \exists a \in A . (b = f(a)) \right\} </annotation></semantics></math></div> <p>which is manifestly the naive definition of <a class="existingWikiWord" href="/nlab/show/image">image</a>.</p> </div> <h3 id="compatibility_with_limits">Compatibility with limits</h3> <div class="num_prop" id="nImagePreservesProducts"> <h6 id="proposition_2">Proposition</h6> <p>In an <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><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, the <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>-image of a product is the product 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>-images:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mi>n</mi></msub><mrow><mo>(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover><mo stretchy="false">)</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub><mo>)</mo></mrow><mo>≃</mo><mrow><mo>(</mo><msub><mi>im</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>im</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> im_n\left(X_1 \times X_2 \stackrel{(f,g)}{\longrightarrow}) X_2 \times Y_2\right) \simeq \left( im_n(f) \times im_n(g) \longrightarrow X_2 \times Y_2 \right) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g)</annotation></semantics></math> is the product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,id)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g,id)</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_2 \times Y_2</annotation></semantics></math>, namely there is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> 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> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && X_1 \times Y_1 \\ & {}^{\mathllap{(id,g)}}\swarrow && \searrow^{\mathrlap{(f,id)}} \\ X_1 \times Y_2 && \swArrow_{\simeq} && X_2 \times Y_1 \\ & {}_{\mathllap{(f,id)}}\searrow && \swarrow_{\mathrlap{(id,g)}} \\ && X_2 \times Y_2 } </annotation></semantics></math></div> <p>But by <a href="n-truncated+object+of+an+infinity-category#nTruncationInToposPreservesFiniteProducts">this proposition</a>, <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 the slice <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>-topos preserves products, so that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>im</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msub><mi>τ</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>τ</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>im</mi> <mi>n</mi></msub><mi>f</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>Y</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>im</mi> <mi>n</mi></msub><mi>g</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>im</mi> <mi>n</mi></msub><mi>f</mi><mo>,</mo><msub><mi>im</mi> <mi>n</mi></msub><mi>g</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} im_n(f,g) & = \tau_n (f,g) \\ & \simeq \tau_n ((f,id)\times_{X_2 \times Y_1} (id,g)) \\ &\simeq (\tau_n(f,id)) \times_{X_2 \times Y_2} (\tau_n(id,g)) \\ & \simeq (im_n f, id)\times_{X_2 \times Y_2} (id, im_n g) \\ & \simeq (im_n f, im_n g) \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The <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>-image of the <a class="existingWikiWord" href="/nlab/show/looping">looping</a> of a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</a> is the looping of its <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>-image</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>Ω</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Ω</mi><mo stretchy="false">(</mo><msub><mi>im</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> im_n (\Omega(f)) \simeq \Omega(im_{n+1}(f)) \,. </annotation></semantics></math></div> <p>i.e. we have the following diagram, where the columns are <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</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>Ω</mi><mi>f</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>Ω</mi><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>im</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>Ω</mi><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Ω</mi><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mo>*</mo></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>im</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>≃</mo><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>f</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mi>im</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega f \colon & \Omega X &\longrightarrow& im_n(\Omega f) &\longrightarrow& \Omega Y \\ & \downarrow && \downarrow && \downarrow \\ id_\ast \colon & \ast &\longrightarrow & im_{n+1}(id_\ast) \simeq \ast &\longrightarrow& \ast \\ & \downarrow && \downarrow && \downarrow \\ f\colon & X &\stackrel{}{\longrightarrow}& im_{n+1}(f) &\stackrel{}{\longrightarrow}& Y } </annotation></semantics></math></div></div> <h2 id="examples">Examples</h2> <h3 id="automorphisms">Automorphisms</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">V \in \mathbf{H}</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/compact+object+in+an+%28%E2%88%9E%2C1%29-category">small object</a>, and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mover><mo>→</mo><mrow><mo>⊢</mo><mi>V</mi></mrow></mover><msub><mi>Obj</mi> <mi>κ</mi></msub></mrow><annotation encoding="application/x-tex"> * \stackrel{\vdash V}{\to} Obj_\kappa </annotation></semantics></math></div> <p>the corresponding morphism to the <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a>. Then the 1-image of this is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbf{Aut}(V)</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the internal <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</p> <h3 id="NImagesOf1FunctorsBetweenGroupoids">Of functors between groupoids</h3> <p>The simplest nontivial case of higher images after the ordinary case of images of functions of sets is images of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> hence betwee <a class="existingWikiWord" href="/nlab/show/1-truncated">1-truncated</a> objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>Grpd</mi><mo>↪</mo></mrow><annotation encoding="application/x-tex">X, Y \in Grpd \hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>.</p> <div class="num_prop" id="FactorizationOf1FunctorsBetween1Groupoids"> <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>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> between <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>, its image factorization is</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 lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msub><mi>im</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> f \colon X \to im_2(f) \to im_1(f) \to Y \,, </annotation></semantics></math></div> <p>where (up to <a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a>)</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">im_1(f) \to Y</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subgroupoid</a> of <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> on those objects <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> such that there is an object <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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f(x) \simeq y</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im_2(f)</annotation></semantics></math> is the groupoid whose objecs are 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> and whose morphisms are equivalence classes of morphisms 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> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha,\beta \in Mor(X)</annotation></semantics></math> are equivalent if they have the same domain and codomain 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> and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\alpha) = f(\beta)</annotation></semantics></math></p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im_2(f)\to im_1(f)</annotation></semantics></math> is the identity on objects and the canonical inclusion on sets of morphisms;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>im</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to im_2(f)</annotation></semantics></math> is the identity on objects and the defining <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> map on sets of morphisms.</p> </li> </ul> </li> </ul> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Evidently <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">im_1(f) \to Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a> (<a class="existingWikiWord" href="/nlab/show/isofibration">isofibration</a>) and so the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> over every point is given by the 1-categorical <a class="existingWikiWord" href="/nlab/show/pullback">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><msub><mi>F</mi> <mi>y</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mi>y</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ F_y &\to& im_1(y) \\ \downarrow && \downarrow \\ * &\stackrel{y}{\to}& Y } \,. </annotation></semantics></math></div> <p>If there is no <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f(x) \simeq y</annotation></semantics></math> then fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">F_y</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a>. If there is such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> then the fiber is the groupoid with that one object and all morphisms 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> on that object which are mapped to the identity morphism, which by construction is only the identity morphism itself, hence the fiber is the point. Hence indeed the homotopy fibers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">im_1(f) \to Y</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/%28-1%29-truncated">(-1)-truncated</a> objects and the map from homotopy fiber 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> to those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">im_1(f)</annotation></semantics></math> is their (-1)-truncation.</p> <p>Next, the <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">im_2(f) \to Y</annotation></semantics></math> over a point <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 the groupoids whose objects are pairs <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><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, (f(x) \to y))</annotation></semantics></math> and whose morphisms are pairs</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msub><mi>x</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo></mrow></mover></mtd> <mtd><msub><mi>x</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>y</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>y</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ x_1 &\stackrel{[\alpha]}{\to}& x_2 \\ f(x_1) &\stackrel{f(\alpha)}{\to}& f(x_2) \\ \downarrow && \downarrow \\ y &=& y } \right) \,. </annotation></semantics></math></div> <p>Notice first that the above is <a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a>: an automorphism of <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><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,(f(x) \to y))</annotation></semantics></math> is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mover><mi>to</mi><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo></mrow></mover><mi>x</mi></mrow><annotation encoding="application/x-tex">x \stackrel{[\alpha]}{to} x</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>id</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(\alpha) = id_{f(x)}</annotation></semantics></math> and so there is precisely one such, namely the equivalence class of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">id_x</annotation></semantics></math>.</p> <p>Notice second that 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> itself have the same form, only that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\alpha \colon x_1 \to x_2</annotation></semantics></math> appears itself, not as its equivalence class. Also if two objects in the homotopy fiber 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> are connected by a morphism, then by construction so they are in the homotopy fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>im</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">im_2(f) \to Y</annotation></semantics></math> and hence the latter is indeed the <a class="existingWikiWord" href="/nlab/show/0-truncation">0-truncation</a> of the former.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The factroization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msub><mi>im</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to im_2(f) \to im_1(f) \to Y</annotation></semantics></math> of prop. <a class="maruku-ref" href="#FactorizationOf1FunctorsBetween1Groupoids"></a> exhibits a <em><a class="existingWikiWord" href="/nlab/show/ternary+factorization+system">ternary factorization system</a></em> in <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>This situation can also be considered from the perspective of the formalization of <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">stuff, structure, property</a>. See there at <em><a href="stuff%2C+structure%2C+property#AFactorizationSystem">A factorization system</a></em>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/image">image</a>, <a class="existingWikiWord" href="/nlab/show/coimage">coimage</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+image">homotopy image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+system">Postnikov system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-ary+factorization+system">k-ary factorization system</a></p> </li> </ul> <h2 id="references">References</h2> <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>-image factorization in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is in</p> <ul> <li><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></li> </ul> <p>A construction 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>-image factorizations in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> using only <a class="existingWikiWord" href="/nlab/show/homotopy+pushouts">homotopy pushouts</a> and specifically <a class="existingWikiWord" href="/nlab/show/join+of+topological+spaces">joins</a> (instead of more general <a class="existingWikiWord" href="/nlab/show/higher+inductive+types">higher inductive types</a>) is described in</p> <ul> <li id="Rijke17"><a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, <em>The join construction</em> (<a href="https://arxiv.org/abs/1701.07538">arXiv:1701.07538</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 29, 2024 at 16:15:27. 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