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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">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </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</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></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='#pushouts_in_'>Pushouts in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math></a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#in_any_category'>In any category</a></li> <li><a href='#in_a_quasitopos'>In a quasitopos</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>pushout</strong> is an ubiquitous construction in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> providing a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> for the diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo><mo>←</mo><mo>•</mo><mo>→</mo><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet\leftarrow\bullet\rightarrow\bullet</annotation></semantics></math>. It is dual to the notion of a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>.</p> <h2 id="pushouts_in_">Pushouts in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math></h2> <p>In the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> a ‘pushout’ is a quotient of the disjoint union of two sets. Given a diagram of sets and functions like this:</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="185.364" height="84.441" viewBox="0 0 185.364 84.441"> <defs> <g> <g id="IQ6epd49xap8noFD-ZlHxicTu28=-glyph-0-0"> </g> <g id="IQ6epd49xap8noFD-ZlHxicTu28=-glyph-0-1"> <path d="M 11.15625 -10.375 C 11.15625 -10.515625 11.046875 -10.515625 11.03125 -10.515625 C 11 -10.515625 10.9375 -10.515625 10.8125 -10.359375 L 9.78125 -9.109375 C 9.265625 -10.015625 8.4375 -10.515625 7.3125 -10.515625 C 4.09375 -10.515625 0.75 -7.25 0.75 -3.734375 C 0.75 -1.234375 2.5 0.3125 4.671875 0.3125 C 5.875 0.3125 6.921875 -0.1875 7.78125 -0.921875 C 9.078125 -2.015625 9.46875 -3.46875 9.46875 -3.578125 C 9.46875 -3.71875 9.34375 -3.71875 9.3125 -3.71875 C 9.171875 -3.71875 9.15625 -3.625 9.125 -3.5625 C 8.4375 -1.234375 6.421875 -0.125 4.921875 -0.125 C 3.34375 -0.125 1.96875 -1.140625 1.96875 -3.25 C 1.96875 -3.734375 2.125 -6.328125 3.8125 -8.296875 C 4.625 -9.25 6.03125 -10.078125 7.453125 -10.078125 C 9.09375 -10.078125 9.828125 -8.71875 9.828125 -7.203125 C 9.828125 -6.8125 9.78125 -6.484375 9.78125 -6.421875 C 9.78125 -6.28125 9.9375 -6.28125 9.984375 -6.28125 C 10.140625 -6.28125 10.15625 -6.296875 10.21875 -6.578125 Z M 11.15625 -10.375 "></path> </g> <g id="IQ6epd49xap8noFD-ZlHxicTu28=-glyph-0-2"> <path d="M 2.546875 -1.65625 C 2.015625 -0.78125 1.515625 -0.484375 0.796875 -0.4375 C 0.625 -0.421875 0.515625 -0.421875 0.515625 -0.15625 C 0.515625 -0.0625 0.578125 0 0.6875 0 C 0.953125 0 1.625 -0.03125 1.890625 -0.03125 C 2.328125 -0.03125 2.8125 0 3.21875 0 C 3.3125 0 3.5 0 3.5 -0.28125 C 3.5 -0.421875 3.375 -0.4375 3.28125 -0.4375 C 2.9375 -0.46875 2.65625 -0.578125 2.65625 -0.9375 C 2.65625 -1.15625 2.75 -1.3125 2.9375 -1.640625 L 4.078125 -3.53125 L 7.890625 -3.53125 C 7.90625 -3.390625 7.90625 -3.265625 7.921875 -3.140625 C 7.96875 -2.75 8.140625 -1.1875 8.140625 -0.90625 C 8.140625 -0.46875 7.375 -0.4375 7.140625 -0.4375 C 6.96875 -0.4375 6.8125 -0.4375 6.8125 -0.171875 C 6.8125 0 6.953125 0 7.03125 0 C 7.296875 0 7.59375 -0.03125 7.84375 -0.03125 L 8.6875 -0.03125 C 9.609375 -0.03125 10.265625 0 10.28125 0 C 10.375 0 10.546875 0 10.546875 -0.28125 C 10.546875 -0.4375 10.40625 -0.4375 10.1875 -0.4375 C 9.359375 -0.4375 9.34375 -0.5625 9.3125 -1.015625 L 8.390625 -10.34375 C 8.359375 -10.640625 8.3125 -10.671875 8.140625 -10.671875 C 7.984375 -10.671875 7.90625 -10.640625 7.765625 -10.40625 Z M 4.328125 -3.953125 L 7.328125 -8.984375 L 7.84375 -3.953125 Z M 4.328125 -3.953125 "></path> </g> <g id="IQ6epd49xap8noFD-ZlHxicTu28=-glyph-0-3"> <path d="M 5.46875 -9.1875 C 5.59375 -9.734375 5.65625 -9.765625 6.25 -9.765625 L 8.1875 -9.765625 C 9.875 -9.765625 9.875 -8.328125 9.875 -8.203125 C 9.875 -6.984375 8.65625 -5.453125 6.6875 -5.453125 L 4.546875 -5.453125 Z M 7.984375 -5.328125 C 9.625 -5.625 11.09375 -6.765625 11.09375 -8.140625 C 11.09375 -9.3125 10.0625 -10.203125 8.375 -10.203125 L 3.578125 -10.203125 C 3.296875 -10.203125 3.171875 -10.203125 3.171875 -9.921875 C 3.171875 -9.765625 3.296875 -9.765625 3.53125 -9.765625 C 4.4375 -9.765625 4.4375 -9.65625 4.4375 -9.484375 C 4.4375 -9.453125 4.4375 -9.359375 4.375 -9.140625 L 2.359375 -1.109375 C 2.21875 -0.578125 2.203125 -0.4375 1.15625 -0.4375 C 0.859375 -0.4375 0.71875 -0.4375 0.71875 -0.171875 C 0.71875 0 0.8125 0 1.109375 0 L 6.234375 0 C 8.515625 0 10.28125 -1.734375 10.28125 -3.234375 C 10.28125 -4.46875 9.203125 -5.21875 7.984375 -5.328125 Z M 5.875 -0.4375 L 3.859375 -0.4375 C 3.640625 -0.4375 3.609375 -0.4375 3.53125 -0.453125 C 3.359375 -0.46875 3.34375 -0.5 3.34375 -0.609375 C 3.34375 -0.71875 3.375 -0.8125 3.40625 -0.9375 L 4.453125 -5.15625 L 7.265625 -5.15625 C 9.015625 -5.15625 9.015625 -3.515625 9.015625 -3.390625 C 9.015625 -1.953125 7.71875 -0.4375 5.875 -0.4375 Z M 5.875 -0.4375 "></path> </g> <g id="IQ6epd49xap8noFD-ZlHxicTu28=-glyph-1-0"> </g> <g id="IQ6epd49xap8noFD-ZlHxicTu28=-glyph-1-1"> <path d="M 3.8125 -3.96875 L 4.75 -3.96875 C 4.9375 -3.96875 5.0625 -3.96875 5.0625 -4.15625 C 5.0625 -4.296875 4.9375 -4.296875 4.765625 -4.296875 L 3.875 -4.296875 C 4.03125 -5.1875 4.140625 -5.765625 4.234375 -6.203125 C 4.28125 -6.375 4.3125 -6.484375 4.453125 -6.609375 C 4.578125 -6.71875 4.671875 -6.734375 4.78125 -6.734375 C 4.921875 -6.734375 5.078125 -6.703125 5.21875 -6.625 C 5.15625 -6.609375 5.109375 -6.578125 5.046875 -6.546875 C 4.890625 -6.453125 4.765625 -6.28125 4.765625 -6.078125 C 4.765625 -5.859375 4.9375 -5.71875 5.15625 -5.71875 C 5.453125 -5.71875 5.71875 -5.96875 5.71875 -6.3125 C 5.71875 -6.78125 5.25 -7.015625 4.765625 -7.015625 C 4.421875 -7.015625 3.796875 -6.859375 3.484375 -5.9375 C 3.390625 -5.71875 3.390625 -5.6875 3.125 -4.296875 L 2.375 -4.296875 C 2.171875 -4.296875 2.046875 -4.296875 2.046875 -4.109375 C 2.046875 -3.96875 2.1875 -3.96875 2.359375 -3.96875 L 3.0625 -3.96875 L 2.34375 -0.09375 C 2.15625 0.90625 2 1.75 1.46875 1.75 C 1.453125 1.75 1.234375 1.75 1.046875 1.640625 C 1.5 1.53125 1.5 1.109375 1.5 1.09375 C 1.5 0.875 1.328125 0.734375 1.109375 0.734375 C 0.84375 0.734375 0.546875 0.953125 0.546875 1.328125 C 0.546875 1.75 0.984375 2.03125 1.46875 2.03125 C 2.078125 2.03125 2.5 1.390625 2.625 1.140625 C 2.984375 0.484375 3.21875 -0.75 3.234375 -0.859375 Z M 3.8125 -3.96875 "></path> </g> <g id="IQ6epd49xap8noFD-ZlHxicTu28=-glyph-1-2"> <path d="M 4.9375 -3.671875 C 4.984375 -3.828125 4.984375 -3.890625 4.984375 -3.890625 C 4.984375 -4.125 4.796875 -4.203125 4.671875 -4.203125 C 4.4375 -4.203125 4.25 -4.03125 4.21875 -3.8125 C 4.140625 -3.96875 3.84375 -4.390625 3.234375 -4.390625 C 2.046875 -4.390625 0.75 -3.078125 0.75 -1.609375 C 0.75 -0.53125 1.46875 0 2.203125 0 C 2.65625 0 3.078125 -0.25 3.40625 -0.53125 L 3.1875 0.359375 C 3.078125 0.78125 3 1.0625 2.609375 1.390625 C 2.1875 1.75 1.8125 1.75 1.5625 1.75 C 1.3125 1.75 1.078125 1.75 0.84375 1.6875 C 1.0625 1.578125 1.15625 1.359375 1.15625 1.203125 C 1.15625 0.953125 0.984375 0.828125 0.765625 0.828125 C 0.515625 0.828125 0.203125 1.03125 0.203125 1.421875 C 0.203125 2 0.984375 2.03125 1.578125 2.03125 C 3 2.03125 3.703125 1.28125 3.859375 0.671875 Z M 3.59375 -1.3125 C 3.53125 -1.03125 3.3125 -0.84375 3.09375 -0.640625 C 3.015625 -0.578125 2.625 -0.28125 2.21875 -0.28125 C 1.828125 -0.28125 1.53125 -0.609375 1.53125 -1.203125 C 1.53125 -1.625 1.78125 -2.71875 2.046875 -3.21875 C 2.375 -3.78125 2.84375 -4.109375 3.234375 -4.109375 C 3.90625 -4.109375 4.09375 -3.375 4.09375 -3.296875 L 4.0625 -3.15625 Z M 3.59375 -1.3125 "></path> </g> </g> </defs> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#IQ6epd49xap8noFD-ZlHxicTu28=-glyph-0-1" x="86.46" y="15.68"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#IQ6epd49xap8noFD-ZlHxicTu28=-glyph-0-2" x="6.434" y="78.972"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#IQ6epd49xap8noFD-ZlHxicTu28=-glyph-0-3" x="167.06" y="78.972"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -12.443719 20.821563 L -68.221062 -23.139375 " transform="matrix(1, 0, 0, -1, 92.682, 42.22)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.487453 2.870725 C -2.030201 1.147462 -1.018873 0.335431 -0.0000179234 -0.0000590898 C -1.020114 -0.335076 -2.033697 -1.147752 -2.486106 -2.870288 " transform="matrix(-0.78534, 0.61899, 0.61899, 0.78534, 24.27346, 65.50787)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#IQ6epd49xap8noFD-ZlHxicTu28=-glyph-1-1" x="42.463" y="38.074"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 11.607063 20.821563 L 67.318 -22.834687 " transform="matrix(1, 0, 0, -1, 92.682, 42.22)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.48808 2.868766 C -2.031989 1.146735 -1.020059 0.336697 -0.000931965 -0.00119402 C -1.020712 -0.333798 -2.031902 -1.146028 -2.488529 -2.868627 " transform="matrix(0.7871, 0.61678, 0.61678, -0.7871, 160.18897, 65.20276)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#IQ6epd49xap8noFD-ZlHxicTu28=-glyph-1-2" x="135.848" y="37.921"></use> </g> </svg> </div> <p>the ‘pushout’ of this diagram is the set <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> obtained by taking the disjoint union <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A + B</annotation></semantics></math> and identifying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math> if there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">x \in C</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>a</mi></mrow><annotation encoding="application/x-tex">f(x) = a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">g(x) = b</annotation></semantics></math> (and all identifications that follow to keep <a class="existingWikiWord" href="/nlab/show/equality">equality</a> an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a>).</p> <p>This construction comes up, for example, when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the intersection of the sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <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>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> are the obvious inclusions. Then the pushout is just the union of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <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>.</p> <p>Note that there are maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i_A : A \to X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>B</mi></msub><mo>:</mo><mi>B</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i_B : B \to X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>a</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i_A(a) = [a]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i_B(b) = [b]</annotation></semantics></math> respectively. These maps make this square commute:</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="187.141" height="147.733" viewBox="0 0 187.141 147.733"> <defs> <g> <g id="UoV2Yx5oVHFUWJLjlVHyhOg74wA=-glyph-0-0"> </g> <g id="UoV2Yx5oVHFUWJLjlVHyhOg74wA=-glyph-0-1"> <path d="M 11.15625 -10.375 C 11.15625 -10.515625 11.046875 -10.515625 11.03125 -10.515625 C 11 -10.515625 10.9375 -10.515625 10.8125 -10.359375 L 9.78125 -9.109375 C 9.265625 -10.015625 8.4375 -10.515625 7.3125 -10.515625 C 4.09375 -10.515625 0.75 -7.25 0.75 -3.734375 C 0.75 -1.234375 2.5 0.3125 4.671875 0.3125 C 5.875 0.3125 6.921875 -0.1875 7.78125 -0.921875 C 9.078125 -2.015625 9.46875 -3.46875 9.46875 -3.578125 C 9.46875 -3.71875 9.34375 -3.71875 9.3125 -3.71875 C 9.171875 -3.71875 9.15625 -3.625 9.125 -3.5625 C 8.4375 -1.234375 6.421875 -0.125 4.921875 -0.125 C 3.34375 -0.125 1.96875 -1.140625 1.96875 -3.25 C 1.96875 -3.734375 2.125 -6.328125 3.8125 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<path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.486371 2.870832 C -2.031595 1.14926 -1.020896 0.336203 -0.00105519 0.000424074 C -1.022177 -0.334983 -2.03291 -1.14798 -2.486069 -2.867637 " transform="matrix(-0.79034, 0.6126, 0.6126, 0.79034, 106.65125, 128.04328)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#UoV2Yx5oVHFUWJLjlVHyhOg74wA=-glyph-1-3" x="138.038" y="116.542"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#UoV2Yx5oVHFUWJLjlVHyhOg74wA=-glyph-2-2" x="141.64175" y="118.21575"></use> </g> </svg> </div> <p>In fact, the pushout is the <a class="existingWikiWord" href="/nlab/show/universal+property">universal</a> solution to finding a <a class="existingWikiWord" href="/nlab/show/commutative+square">commutative square</a> like this. 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y="116.774"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#vEMJmf2gwAmTpAi4Ka3Zr_RziLA=-glyph-2-1" x="40.80825" y="118.449"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 68.335188 -10.824406 L 12.479719 -54.543156 " transform="matrix(1, 0, 0, -1, 92.782, 73.867)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.487963 2.868204 C -2.033422 1.148287 -1.018694 0.334237 -0.00113634 -0.000853555 C -1.020731 -0.333846 -2.034595 -1.147538 -2.484207 -2.869153 " transform="matrix(-0.78746, 0.61633, 0.61633, 0.78746, 105.07385, 128.55606)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#vEMJmf2gwAmTpAi4Ka3Zr_RziLA=-glyph-1-3" x="136.517" y="116.799"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#vEMJmf2gwAmTpAi4Ka3Zr_RziLA=-glyph-2-2" x="140.79825" y="118.47275"></use> </g> </svg> </div> <p>there is a unique function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">h: X \to Y</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><msub><mi>i</mi> <mi>A</mi></msub><mo>=</mo><msub><mi>j</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex"> h i_A = j_A </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><msub><mi>i</mi> <mi>B</mi></msub><mo>=</mo><msub><mi>j</mi> <mi>B</mi></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> h i_B = j_B .</annotation></semantics></math></div> <p>Since this universal property expresses the concept of pushout purely arrow-theoretically, we can formulate it in any category. It is, in fact, a simple special case of a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>.</p> <h2 id="definition">Definition</h2> <p>A <strong>pushout</strong> is a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> like this:</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="171.183" height="80.83" viewBox="0 0 171.183 80.83"> <defs> <g> <g id="ClkiBvWGJLX9f7tPQ517mHw63tw=-glyph-0-0"> </g> <g id="ClkiBvWGJLX9f7tPQ517mHw63tw=-glyph-0-1"> <path d="M 5.84375 -5.609375 C 5.5625 -5.609375 5.421875 -5.609375 5.21875 -5.4375 C 5.125 -5.359375 4.953125 -5.140625 4.953125 -4.90625 C 4.953125 -4.59375 5.1875 -4.421875 5.46875 -4.421875 C 5.828125 -4.421875 6.234375 -4.71875 6.234375 -5.3125 C 6.234375 -6.03125 5.546875 -6.59375 4.515625 -6.59375 C 2.546875 -6.59375 0.59375 -4.453125 0.59375 -2.328125 C 0.59375 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class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is also called the <strong>pushout</strong>. It has the universal property already described above in the special case of the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>.</p> <p>Other terms: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is a <strong>cofibred coproduct</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <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>, or (especially in <a class="existingWikiWord" href="/nlab/show/algebraic+categories">algebraic categories</a> when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>) a free product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> <strong><a class="existingWikiWord" href="/nlab/show/amalgamation">amalgamated</a> sum</strong> (<a href="#GabrielZisman67">Gabriel & Zisman (1967), p. 1</a>) or more simply an <strong><a class="existingWikiWord" href="/nlab/show/amalgamation">amalgamation</a></strong> (or <strong>amalgam</strong>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <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>.</p> <p>The concept of pushout is a special case of the notion of <strong><a class="existingWikiWord" href="/nlab/show/cofiber+coproduct">wide pushout</a></strong> (compare <a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a>), where one takes the colimit of a diagram which consists of a set of arrows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><mi>c</mi><mo>→</mo><msub><mi>a</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{f_i: c \to a_i\}_{i \in I}</annotation></semantics></math>. Thus an ordinary pushout is the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> has cardinality <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>.</p> <p>Note that the concept of pushout is dual to the concept of <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>: that is, a pushout in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the same as a pullback in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math>.</p> <p>See <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> for more details.</p> <h2 id="properties">Properties</h2> <h3 id="in_any_category">In any category</h3> <div class="num_prop" id="PushoutsAsCoequalizers"> <h6 id="proposition">Proposition</h6> <p><strong>(pushouts as coequalizers)</strong></p> <p>If <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> exist in some <a class="existingWikiWord" href="/nlab/show/category">category</a>, then the pushout</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="120.583" height="98.712" viewBox="0 0 120.583 98.712"> <defs> <g> <g id="Nins_coO5cL0jPjz1kqbgG2NXDs=-glyph-0-0"> </g> <g id="Nins_coO5cL0jPjz1kqbgG2NXDs=-glyph-0-1"> <path d="M 4.5 -1.78125 C 4.421875 -1.53125 4.421875 -1.5 4.21875 -1.203125 C 3.890625 -0.796875 3.21875 -0.15625 2.53125 -0.15625 C 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display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> into the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>.</p> </div> <div class="num_prop" id="PushoutPreservesEpimorphisms"> <h6 id="proposition_2">Proposition</h6> <p><strong>(pushouts preserves epimorphisms and isomorphisms)</strong></p> <p>Pushouts preserve <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a> and <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>:</p> <p>If</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="111.437" height="88.251" viewBox="0 0 111.437 88.251"> <defs> <g> <g id="e3OB0wqXOI5zwrbO0DEZS2_CCtU=-glyph-0-0"> </g> <g id="e3OB0wqXOI5zwrbO0DEZS2_CCtU=-glyph-0-1"> <path d="M 4.5 -1.78125 C 4.421875 -1.53125 4.421875 -1.5 4.21875 -1.203125 C 3.890625 -0.796875 3.21875 -0.15625 2.53125 -0.15625 C 1.90625 -0.15625 1.5625 -0.703125 1.5625 -1.578125 C 1.5625 -2.40625 2.03125 -4.078125 2.3125 -4.703125 C 2.828125 -5.75 3.53125 -6.28125 4.109375 -6.28125 C 5.09375 -6.28125 5.28125 -5.0625 5.28125 -4.9375 C 5.28125 -4.921875 5.25 -4.734375 5.234375 -4.703125 Z M 5.453125 -5.59375 C 5.28125 -5.984375 4.890625 -6.59375 4.109375 -6.59375 C 2.421875 -6.59375 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class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">f_\ast g</annotation></semantics></math> is an epimorphism;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">f_\ast g</annotation></semantics></math> is an isomorphism.</p> </li> </ol> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/pasting+law+for+pushouts">pasting law for pushouts</a>)</strong></p> <p>Consider a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the following shape in any category:</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="177.843" height="79.792" viewBox="0 0 177.843 79.792"> <defs> <g> <g id="i_Rs3Nwg35jWnC6q8BeO44l1Ag4=-glyph-0-0"> </g> <g id="i_Rs3Nwg35jWnC6q8BeO44l1Ag4=-glyph-0-1"> <path d="M 7.078125 -6.09375 C 6.609375 -6 6.421875 -5.640625 6.421875 -5.359375 C 6.421875 -5 6.703125 -4.890625 6.921875 -4.890625 C 7.359375 -4.890625 7.671875 -5.265625 7.671875 -5.671875 C 7.671875 -6.296875 6.953125 -6.59375 6.328125 -6.59375 C 5.421875 -6.59375 4.921875 -5.6875 4.78125 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transform="matrix(0, 1, 1, 0, 165.97965, 61.93181)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -67.476687 -30.689937 L -12.2345 -30.689937 " transform="matrix(1, 0, 0, -1, 88.922, 39.896)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.486024 2.86863 C -2.032899 1.145974 -1.02118 0.333474 -0.00164875 0.0014425 C -1.02118 -0.334495 -2.032899 -1.146995 -2.486024 -2.869651 " transform="matrix(1, 0, 0, -1, 76.92743, 70.58738)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 9.222531 -30.689937 L 64.464719 -30.689937 " transform="matrix(1, 0, 0, -1, 88.922, 39.896)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.485445 2.86863 C -2.03232 1.145974 -1.020601 0.333474 -0.00107 0.0014425 C -1.020601 -0.334495 -2.03232 -1.146995 -2.485445 -2.869651 " transform="matrix(1, 0, 0, -1, 153.62607, 70.58738)"></path> </svg> </div> <p>If the left square is a <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, then the total rectangle is a pushout if and only if the right square is a pushout.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>See the proof of the dual property for <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>The converse implication does not hold: it may happen that the outer and the right square are pushouts, but not the left square.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>See the proof of the dual proposition for <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>.</p> </div> <h3 id="in_a_quasitopos">In a quasitopos</h3> <div class="num_prop" id="PushoutOfStrongMonomorphismInQuasitopos"> <h6 id="proposition_5">Proposition</h6> <p><strong>pushout of <a class="existingWikiWord" href="/nlab/show/strong+monomorphism">strong monomorphism</a> in <a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></strong></p> <p>Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">T</mi><mo>,</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathrm{T},\mathcal{C})</annotation></semantics></math> is either</p> <ul> <li>(<a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>,<a class="existingWikiWord" href="/nlab/show/topos">topos</a>), or</li> <li>(<a class="existingWikiWord" href="/nlab/show/strong+monomorphism">strong monomorphism</a>,<a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a>)</li> </ul> <p>Suppose that</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="132.588" height="92.799" viewBox="0 0 132.588 92.799"> <defs> <g> <g id="qm6YHsYCBeZ1dcNhta3gkDkzcdA=-glyph-0-0"> </g> <g id="qm6YHsYCBeZ1dcNhta3gkDkzcdA=-glyph-0-1"> <path d="M 10.84375 -6.546875 C 10.84375 -9 9.234375 -10.515625 7.140625 -10.515625 C 3.9375 -10.515625 0.71875 -7.078125 0.71875 -3.625 C 0.71875 -1.28125 2.265625 0.3125 4.4375 0.3125 C 7.578125 0.3125 10.84375 -2.953125 10.84375 -6.546875 Z M 4.53125 -0.03125 C 3.296875 -0.03125 2 -0.921875 2 -3.25 C 2 -4.609375 2.5 -6.84375 3.71875 -8.328125 C 4.8125 -9.65625 6.0625 -10.1875 7.0625 -10.1875 C 8.375 -10.1875 9.65625 -9.234375 9.65625 -7.078125 C 9.65625 -5.75 9.078125 -3.671875 8.078125 -2.25 C 6.984375 -0.734375 5.625 -0.03125 4.53125 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stroke-opacity="1" stroke-miterlimit="10" d="M 47.194281 20.821875 L 47.194281 -20.342187 " transform="matrix(1, 0, 0, -1, 66.294, 46.4)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.486806 2.870295 C -2.033681 1.147639 -1.018056 0.335139 0.001475 -0.00079875 C -1.018056 -0.33283 -2.033681 -1.149236 -2.486806 -2.867986 " transform="matrix(0, 1, 1, 0, 113.48908, 66.9829)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#qm6YHsYCBeZ1dcNhta3gkDkzcdA=-glyph-2-3" x="117.004" y="49.859"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -27.844781 -33.017969 L 27.366156 -33.017969 " transform="matrix(1, 0, 0, -1, 66.294, 46.4)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.486488 2.869145 C -2.033363 1.146489 -1.021644 0.333989 0.00179375 -0.00194875 C -1.021644 -0.33398 -2.033363 -1.14648 -2.486488 -2.869136 " transform="matrix(1, 0, 0, -1, 93.90055, 79.41602)"></path> </svg> </div> <p>is a <a class="existingWikiWord" href="/nlab/show/commutative+diagram">commutative diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> such that</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">T</mi></mrow><annotation encoding="application/x-tex">\mathrm{T}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></li> <li>the diagram is a pushout in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></li> </ul> <p>Then</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">T</mi></mrow><annotation encoding="application/x-tex">\mathrm{T}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></li> <li>the diagram is a pullback in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></li> </ul> </div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></em> <a href="quasitopos#PushoutOfStrongMonos">this lemma</a>. Note that the result for quasitoposes immediately implies the result for toposes, since all monomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i: A \to B</annotation></semantics></math> in a topos are <a class="existingWikiWord" href="/nlab/show/regular+monomorphism">regular</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> being the equalizer of the arrows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>i</mi></msub><mo>,</mo><mi>t</mi><mo>∘</mo><mo>!</mo><mo>:</mo><mi>B</mi><mo>→</mo><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\chi_i, t \circ !: B \to 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stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -27.193719 -22.780938 L 29.747687 22.121406 " transform="matrix(1, 0, 0, -1, 52.182, 41.93)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.488515 2.868879 C -2.031663 1.145498 -1.02052 0.333225 0.000682103 0.000562173 C -1.019499 -0.334218 -2.033278 -1.146662 -2.486094 -2.869102 " transform="matrix(0.78519, -0.61917, -0.61917, -0.78519, 82.117, 19.66102)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#e2NtYN8cidCmIFGGRGie30-qD3o=-glyph-3-1" x="47.19" y="38.597"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -27.193719 -32.72625 L 28.017219 -32.72625 " transform="matrix(1, 0, 0, -1, 52.182, 41.93)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.486655 2.870274 C -2.03353 1.147617 -1.021811 0.335117 0.00162625 -0.00082 C -1.021811 -0.332851 -2.03353 -1.149258 -2.486655 -2.868008 " transform="matrix(1, 0, 0, -1, 80.43978, 74.65543)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#e2NtYN8cidCmIFGGRGie30-qD3o=-glyph-2-2" x="47.571" y="82.461"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#e2NtYN8cidCmIFGGRGie30-qD3o=-glyph-4-1" x="54.1435" y="83.97975"></use> </g> </svg> </div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\chi_i</annotation></semantics></math> is the classifying map of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>) and therefore strong.</p> <h2 id="Examples">Examples</h2> <p> <div class='num_remark'> <h6>Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of <a class="existingWikiWord" href="/nlab/show/injections">injections</a> of <a class="existingWikiWord" href="/nlab/show/Sets">Sets</a> is called the <a class="existingWikiWord" href="/nlab/show/union">union</a> of the sets.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>A pushout of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> in <a class="existingWikiWord" href="/nlab/show/Grps">Grps</a> is called their <em><a class="existingWikiWord" href="/nlab/show/amalgamated+free+product">amalgamated free product</a></em></p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>In <a class="existingWikiWord" href="/nlab/show/topology">topology</a>, <a class="existingWikiWord" href="/nlab/show/space+attachments">space attachments</a> are pushouts in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>.</p> </div> </p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/amalgamation+property">amalgamation property</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/wide+pushout">wide pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pushout-product">pushout-product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> </li> </ul> <h2 id="references">References</h2> <p>Early use of the terminology “pushout”:</p> <ul> <li id="Mitchell65"><a class="existingWikiWord" href="/nlab/show/Barry+Mitchell">Barry Mitchell</a>, Section I.7 of: <em>Theory of categories</em>, Pure and Applied Mathematics <strong>17</strong>, Academic Press (1965) [<a href="https://www.elsevier.com/books/theory-of-categories/mitchell/978-0-12-499250-4">ISBN:978-0-12-499250-4</a>]</li> </ul> <p>Early use of the terminology “amalgamated sums”:</p> <ul> <li id="GabrielZisman67"><a class="existingWikiWord" href="/nlab/show/Pierre+Gabriel">Pierre Gabriel</a>, <a class="existingWikiWord" href="/nlab/show/Michel+Zisman">Michel Zisman</a>, p 1 of: <em><a class="existingWikiWord" href="/nlab/show/Calculus+of+fractions+and+homotopy+theory">Calculus of fractions and homotopy theory</a></em>, Ergebnisse der Mathematik und ihrer Grenzgebiete <strong>35</strong>, Springer (1967) [<a href="https://link.springer.com/book/10.1007/978-3-642-85844-4">doi:10.1007/978-3-642-85844-4</a>, <a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf">pdf</a>]</li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, p. 65-66 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Section 2.5 in Vol. 1: <em>Basic Category Theory</em> of: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em>, Encyclopedia of Mathematics and its Applications <strong>50</strong> Cambridge University Press (1994) (<a href="https://doi.org/10.1017/CBO9780511525858">doi:10.1017/CBO9780511525858</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 29, 2023 at 15:36:09. 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