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class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</a></p> </li> </ul> </div></div> <h4 id="limits_and_colimits">Limits and colimits</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/limit">limits and colimits</a></strong></p> <h2 id="1categorical">1-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit and colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">commutativity of limits and colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+limit">small limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+limit">connected limit</a>, <a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved limit</a>, <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected limit</a>, <a class="existingWikiWord" href="/nlab/show/created+limit">created limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product">product</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>, <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>, <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>, <a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, <a class="existingWikiWord" href="/nlab/show/direct+product">direct product</a>, <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end and coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibered+limit">fibered limit</a></p> </li> </ul> <h2 id="2categorical">2-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isoinserter">isoinserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PIE-limit">PIE-limit</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> </ul> <h2 id="1categorical_2">(∞,1)-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id="modelcategorical">Model-categorical</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+product">homotopy product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equalizer">homotopy equalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+totalization">homotopy totalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+end">homotopy end</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coproduct">homotopy coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</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/homotopy+pushout">homotopy pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+realization">homotopy realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-limits+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>An <em>inverter</em> is a particular kind of <a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a> in a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, which universally renders a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> invertible.</p> <h2 id="definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⇉</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f,g\colon A\rightrightarrows B</annotation></semantics></math> be a pair of parallel 1-morphisms in a 2-category and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>f</mi><mo>→</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">\alpha\colon f\to g</annotation></semantics></math> a 2-morphism. The <strong>inverter</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> is a universal object <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> equipped with a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">v\colon V\to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">\alpha v</annotation></semantics></math> is invertible.</p> <p>More precisely, universality means that for any 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>, the induced functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(X, v) : Hom(X,V) \to Hom(X,A)</annotation></semantics></math></div> <p>is fully faithful (so that <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> is <a class="existingWikiWord" href="/nlab/show/representably+fully+faithful">representably fully faithful</a>), and its <a class="existingWikiWord" href="/nlab/show/replete+image">replete image</a> consists precisely of those morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">u\colon X\to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">\alpha u</annotation></semantics></math> is invertible. If the above functor is additionally an <em>isomorphism</em> of categories onto the exact subcategory of such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>, then we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mover><mo>→</mo><mi>v</mi></mover><mi>A</mi></mrow><annotation encoding="application/x-tex">V\xrightarrow{v} A</annotation></semantics></math> is a <strong>strict inverter</strong>.</p> <p>Inverters and strict inverters can be described as a certain sort of <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted</a> 2-limit, where the diagram shape is the <a class="existingWikiWord" href="/nlab/show/walking+structure">walking</a> 2-morphism</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">T=</annotation></semantics></math></p> <div> <svg xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" height="75" se:nonce="47584" width="100"> <g> <title>Layer 1</title> <path d="m20.4,29c12.400003,-21.8 45.6,-22.799997 57.6,-0.6" fill="#ff0000" fill-opacity="0" id="svg_47584_1" marker-end="url(#se_marker_end_svg_47584_1)" stroke="#000000" stroke-width="2"></path> <path d="m20.799999,63c12.400002,-21.799999 45.600002,-22.799999 57.600002,-0.599998" fill="#ff0000" fill-opacity="0" id="svg_47584_2" marker-start="url(#se_marker_start_svg_47584_2)" stroke="#000000" stroke-width="2" transform="rotate(180, 49.5996, 54.5977)"></path> <line fill="none" fill-opacity="0" id="svg_47584_3" marker-end="url(#se_marker_end_svg_47584_3)" stroke="#000000" stroke-width="2" x1="49.400001" x2="49.400001" y1="26.500025" y2="49.700013"></line> <line fill="none" fill-opacity="0" id="svg_47584_4" stroke="#ffffff" stroke-width="5" x1="48.199997" x2="50.203143" y1="23.300317" y2="45.300256"></line> <line fill="none" fill-opacity="0" id="svg_47584_5" stroke="#000000" x1="50.600001" x2="50.600001" y1="25.699876" y2="48.969157"></line> <line fill="none" fill-opacity="0" id="svg_47584_6" stroke="#000000" x1="47.4" x2="47.4" y1="25.699999" y2="46.569115"></line> </g> <defs> <marker id="se_marker_end_svg_47584_1" markerHeight="5" markerUnits="strokeWidth" markerWidth="5" orient="auto" refX="50" refY="50" viewBox="0 0 100 100"> <path d="m100,50l-100,40l30,-40l-30,-40z" fill="#000000" stroke="#000000" stroke-width="10"></path> </marker> <marker id="se_marker_start_svg_47584_2" markerHeight="5" markerUnits="strokeWidth" markerWidth="5" orient="auto" refX="50" refY="50" viewBox="0 0 100 100"> <path d="m0,50l100,40l-30,-40l30,-40z" fill="#000000" stroke="#000000" stroke-width="10"></path> </marker> <marker id="se_marker_end_svg_47584_3" markerHeight="5" markerUnits="strokeWidth" markerWidth="5" orient="auto" refX="50" refY="50" viewBox="0 0 100 100"> <path d="m100,50l-100,40l30,-40l-30,-40z" fill="#000000" stroke="#000000" stroke-width="10"></path> </marker> </defs> </svg><div class="property">category: <a class="category_link" href="/nlab/all_pages/svg">svg</a></div></div> <p>and the weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">T\to Cat</annotation></semantics></math> is 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></mtd> <mtd><mo>→</mo></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mi>Inv</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>→</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ & \to \\ 1 & \Downarrow & Inv\\ & \to } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Inv</mi></mrow><annotation encoding="application/x-tex">Inv</annotation></semantics></math> is the walking <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Inv</mi></mrow><annotation encoding="application/x-tex">Inv</annotation></semantics></math> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>, this weight is equivalent to the terminal weight, and thus an inverter (but not a strict inverter) can also be defined simply as a <a class="existingWikiWord" href="/nlab/show/conical+limit">conical</a> 2-limit over a diagram of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>.</p> <p>An inverter in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">K^{op}</annotation></semantics></math> (see <a class="existingWikiWord" href="/nlab/show/opposite+2-category">opposite 2-category</a>) is called a <strong>coinverter</strong> in <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>. Coinverters in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> are also called <a class="existingWikiWord" href="/nlab/show/localizations">localizations</a>.</p> <h2 id="properties">Properties</h2> <ul> <li> <p>The above description makes it clear that any inverter is in particular a <a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a>.</p> </li> <li> <p>Any strict inverter is, in particular, an inverter. (This is not true for all strict 2-limits.)</p> </li> <li> <p>Inverters can be constructed in a straightforward way from <a class="existingWikiWord" href="/nlab/show/inserters">inserters</a> and <a class="existingWikiWord" href="/nlab/show/equifiers">equifiers</a>: first we insert a 2-morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> going in the opposite direction from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>, then we equify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mi>α</mi></mrow><annotation encoding="application/x-tex">\beta\alpha</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mi>β</mi></mrow><annotation encoding="application/x-tex">\alpha\beta</annotation></semantics></math> with identities. In particular, it follows that strict inverters are <a class="existingWikiWord" href="/nlab/show/PIE-limits">PIE-limits</a>.</p> </li> </ul> <h2 id="references">References</h2> <p>In</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a></em></li> </ul> <p>inverters appear in B1.1.4. The inverter of a transformation between <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>s of <a class="existingWikiWord" href="/nlab/show/topos">topos</a>es is constructed in the proof of corollary 4.1.7 in section B4.1. Coinverters are discussed in section B4.5 there.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on June 1, 2024 at 21:16:32. 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