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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='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#section_Idea'>Idea</a></li> <li><a href='#section_Details'>Details</a></li> <ul> <li><a href='#section_Sketches'>Sketches</a></li> <li><a href='#section_Models_of_sketches'>Models of sketches</a></li> </ul> <li><a href='#section_Examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationToLocallyRepresentableCategories'>Relation to accessible and locally representable categories</a></li> <li><a href='#monoidal_structures_on_the_category_of_sketches'>Monoidal Structures on the Category of Sketches</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#in_enriched_category_theory'>In enriched category theory</a></li> <li><a href='#generalisations'>Generalisations</a></li> </ul> </ul> </div> <p><h2 id='section_Idea'>Idea</h2></p> <p>The notion of a <em>sketch</em> (<a href="#BastianiEhresmann72">Bastiani & Ehresmann 1972</a>) is one formalisation of the notion of a <em><a class="existingWikiWord" href="/nlab/show/theory">theory</a></em>. It is <a class="existingWikiWord" href="/nlab/show/diagram">diagrammatic</a>, and has the advantage of being very close to <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, allowing it to very naturally express the category theoretic structure which is required to construct a <a class="existingWikiWord" href="/nlab/show/model">model</a> of the theory (<a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a>, say). On the other hand, it is not a very concise notion: as Example <a class="maruku-ref" href="#ExampleSketchUnitalMagmas"></a> illustrates, writing down the full details of a sketch even in the simplest examples takes time!</p> <p>There is a precise correspondence between categories of models of sketches and <a class="existingWikiWord" href="/nlab/show/accessible+category">accessible categories</a> and <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable categories</a>, discussed below.</p> <p>The notion of a sketch generalises that of a <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a>. See Example <a class="maruku-ref" href="#ExampleLawvereTheory"></a>.</p> <p><h2 id='section_Details'>Details</h2></p> <p><h3 id='section_Sketches'>Sketches</h3></p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <em>sketch</em> is a <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/category">category</a> <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> equipped with a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/cones">cones</a> and a set <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> of <a class="existingWikiWord" href="/nlab/show/cocones">cocones</a>. Alternatively, it is a <a class="existingWikiWord" href="/nlab/show/directed+graph">directed graph</a> equipped with a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/diagram">diagrams</a>, a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/cones">cones</a>, and a set <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> of <a class="existingWikiWord" href="/nlab/show/cocones">cocones</a>.</p> </div> </p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <em>realized sketch</em> is one where all the cones in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/limit">limit</a> cones and all the cocones 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> are <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> cocones.</p> </div> </p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <em>limit sketch</em> is a sketch with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">C = \emptyset</annotation></semantics></math>.</p> </div> </p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <em>colimit sketch</em> is a sketch with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">L = \emptyset</annotation></semantics></math>.</p> </div> </p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <em>finite product sketch</em> is a limit sketch in which the only cones are those of <a class="existingWikiWord" href="/nlab/show/finite+product">finite product</a> diagrams.</p> </div> </p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <em>finite limit sketch</em> is a limit sketch in which the only cones are those of <a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a> diagrams.</p> </div> </p> <p><h3 id='section_Models_of_sketches'>Models of sketches</h3></p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <em>model of a sketch</em> in a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">T\to \mathcal{C}</annotation></semantics></math> taking each cone in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> to a limit cone and each cocone 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> to a colimit cocone.</p> </div> </p> <p>In particular, <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> is realized if and only if its identity functor is a model.</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>If one takes the definition of a sketch to be that involving directed graphs, a <em>model of a sketch</em> in a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a morphism of directed graphs from the directed graph of the sketch to the underlying directed graph of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, so that diagrams are taken to commutative diagrams, cones are taken to limit cones, and co-cones are taken to colimit cones.</p> </div> </p> <p>Frequently the notion of model is restricted to the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}=Set</annotation></semantics></math>.</p> <p> <div class='num_defn'> <h6>Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/category">category</a> is <em>sketchable</em> if it is the category of models (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>) of a sketch.</p> </div> </p> <p><h2 id='section_Examples'>Examples</h2></p> <p> <div class='num_remark'> <h6>Example</h6> <p>A sketch, more precisely a finite product sketch, for the theory of pointed sets can be constructed as follows. The directed graph can be taken to be the following.</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="103.234" height="18.711" viewBox="0 0 103.234 18.711"> <defs> <g> <g id="USmTuxw3JbTfscMc7jEFo7U3nwY=-glyph-0-0"> </g> <g id="USmTuxw3JbTfscMc7jEFo7U3nwY=-glyph-0-1"> <path d="M 5.46875 -4.46875 C 5.46875 -5.21875 5.078125 -5.265625 4.984375 -5.265625 C 4.703125 -5.265625 4.4375 -4.984375 4.4375 -4.75 C 4.4375 -4.609375 4.515625 -4.515625 4.5625 -4.46875 C 4.6875 -4.359375 5 -4.046875 5 -3.421875 C 5 -2.921875 4.28125 -0.125 2.84375 -0.125 C 2.109375 -0.125 1.96875 -0.734375 1.96875 -1.171875 C 1.96875 -1.765625 2.25 -2.609375 2.5625 -3.46875 C 2.765625 -3.953125 2.8125 -4.078125 2.8125 -4.3125 C 2.8125 -4.8125 2.453125 -5.265625 1.859375 -5.265625 C 0.765625 -5.265625 0.328125 -3.53125 0.328125 -3.4375 C 0.328125 -3.390625 0.375 -3.328125 0.453125 -3.328125 C 0.5625 -3.328125 0.578125 -3.375 0.625 -3.546875 C 0.90625 -4.578125 1.375 -5.03125 1.828125 -5.03125 C 1.9375 -5.03125 2.140625 -5.03125 2.140625 -4.640625 C 2.140625 -4.328125 2.015625 -3.984375 1.828125 -3.53125 C 1.25 -2 1.25 -1.625 1.25 -1.34375 C 1.25 -1.078125 1.296875 -0.578125 1.65625 -0.25 C 2.09375 0.125 2.6875 0.125 2.796875 0.125 C 4.78125 0.125 5.46875 -3.796875 5.46875 -4.46875 Z M 5.46875 -4.46875 "></path> </g> <g id="USmTuxw3JbTfscMc7jEFo7U3nwY=-glyph-1-0"> </g> <g id="USmTuxw3JbTfscMc7jEFo7U3nwY=-glyph-1-1"> <path d="M 2.5 -5.078125 C 2.5 -5.296875 2.484375 -5.296875 2.265625 -5.296875 C 1.9375 -4.984375 1.515625 -4.796875 0.765625 -4.796875 L 0.765625 -4.53125 C 0.984375 -4.53125 1.40625 -4.53125 1.875 -4.734375 L 1.875 -0.65625 C 1.875 -0.359375 1.84375 -0.265625 1.09375 -0.265625 L 0.8125 -0.265625 L 0.8125 0 C 1.140625 -0.03125 1.828125 -0.03125 2.1875 -0.03125 C 2.546875 -0.03125 3.234375 -0.03125 3.5625 0 L 3.5625 -0.265625 L 3.28125 -0.265625 C 2.53125 -0.265625 2.5 -0.359375 2.5 -0.65625 Z M 2.5 -5.078125 "></path> </g> <g id="USmTuxw3JbTfscMc7jEFo7U3nwY=-glyph-1-2"> <path d="M 2.25 -1.625 C 2.375 -1.75 2.703125 -2.015625 2.84375 -2.125 C 3.328125 -2.578125 3.796875 -3.015625 3.796875 -3.734375 C 3.796875 -4.6875 3 -5.296875 2.015625 -5.296875 C 1.046875 -5.296875 0.421875 -4.578125 0.421875 -3.859375 C 0.421875 -3.46875 0.734375 -3.421875 0.84375 -3.421875 C 1.015625 -3.421875 1.265625 -3.53125 1.265625 -3.84375 C 1.265625 -4.25 0.859375 -4.25 0.765625 -4.25 C 1 -4.84375 1.53125 -5.03125 1.921875 -5.03125 C 2.65625 -5.03125 3.046875 -4.40625 3.046875 -3.734375 C 3.046875 -2.90625 2.46875 -2.296875 1.515625 -1.34375 L 0.515625 -0.296875 C 0.421875 -0.21875 0.421875 -0.203125 0.421875 0 L 3.5625 0 L 3.796875 -1.421875 L 3.546875 -1.421875 C 3.53125 -1.265625 3.46875 -0.875 3.375 -0.71875 C 3.328125 -0.65625 2.71875 -0.65625 2.59375 -0.65625 L 1.171875 -0.65625 Z M 2.25 -1.625 "></path> </g> </g> </defs> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" d="M 11.933594 2.835938 C 11.933594 1.269531 10.664062 0 9.097656 0 C 7.53125 0 6.261719 1.269531 6.261719 2.835938 C 6.261719 4.402344 7.53125 5.667969 9.097656 5.667969 C 10.664062 5.667969 11.933594 4.402344 11.933594 2.835938 Z M 11.933594 2.835938 "></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#USmTuxw3JbTfscMc7jEFo7U3nwY=-glyph-0-1" x="3.902" y="13.016"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#USmTuxw3JbTfscMc7jEFo7U3nwY=-glyph-1-1" x="9.561" y="14.809"></use> </g> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" d="M 96.972656 2.835938 C 96.972656 1.269531 95.703125 0 94.136719 0 C 92.570312 0 91.300781 1.269531 91.300781 2.835938 C 91.300781 4.402344 92.570312 5.667969 94.136719 5.667969 C 95.703125 5.667969 96.972656 4.402344 96.972656 2.835938 Z M 96.972656 2.835938 "></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#USmTuxw3JbTfscMc7jEFo7U3nwY=-glyph-0-1" x="88.941" y="13.016"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#USmTuxw3JbTfscMc7jEFo7U3nwY=-glyph-1-2" x="94.6" y="14.809"></use> </g> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 5.852219 -0.0009375 L 78.727219 -0.0009375 " transform="matrix(1, 0, 0, -1, 9.097, 2.835)"></path> <path fill="none" stroke-width="0.31879" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -1.197174 1.592813 C -1.095611 0.995156 -0.00186125 0.100625 0.29892 -0.0009375 C -0.00186125 -0.0985937 -1.095611 -0.997031 -1.197174 -1.594687 " transform="matrix(1, 0, 0, -1, 87.82608, 2.835)"></path> </svg> </div> <p>The set of diagrams can be taken to be empty. The set of cones can be taken to be the set with the single cone given by the vertex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v_{1}</annotation></semantics></math>, i.e. a cone of the empty diagram. The set of co-cones can be taken to be empty.</p> <p>A model of this sketch necessarily sends the vertex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v_{1}</annotation></semantics></math> to a product of the empty diagram, hence to a one element set <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>; sends the vertex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">v_{2}</annotation></semantics></math> to any 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>; and sends the arrow from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v_{1}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">v_{2}</annotation></semantics></math> to an arrow from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, that is, to an element 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>, as required.</p> </div> </p> <p> <div class='num_remark' id='ExampleSketchUnitalMagmas'> <h6>Example</h6> <p>A sketch, more precisely a finite product sketch, for the theory of <a class="existingWikiWord" href="/nlab/show/unital+magmas">unital magmas</a> (<a class="existingWikiWord" href="/nlab/show/sets">sets</a> equipped with a <a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a> which has a two sided <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a>) can be constructed as follows.</p> <p>The directed graph can be taken to be the following:</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="175.748" height="175.748" viewBox="0 0 175.748 175.748"> <defs> <g> 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y="61.702"></use> </g> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 85.040937 79.1865 L 85.040937 6.3115 " transform="matrix(1, 0, 0, -1, -82.205, 87.874)"></path> <path fill="none" stroke-width="0.31879" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -1.195473 1.594267 C -1.097816 0.996611 -0.00016 0.0981737 0.300621 0.0005175 C -0.00016 -0.101045 -1.097816 -0.995576 -1.195473 -1.593233 " transform="matrix(0, 1, 1, 0, 2.83542, 81.56266)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#2GV7fDyU4uLu8LZD4p2Te5X3MSY=-glyph-0-1" x="6.936" y="46.766"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#2GV7fDyU4uLu8LZD4p2Te5X3MSY=-glyph-1-1" x="12.811" y="48.559"></use> </g> <g clip-path="url(#2GV7fDyU4uLu8LZD4p2Te5X3MSY=-clip-3)"> <path fill="none" 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xlink:href="#2GV7fDyU4uLu8LZD4p2Te5X3MSY=-glyph-2-1" x="72.611" y="27.61367"></use> </g> </svg> </div> <p>the fifth consisting of the following</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="90.709" height="90.709" viewBox="0 0 90.709 90.709"> <defs> <g> <g id="UqoBzFCBTzJeN6VVm4-5IAae58Q=-glyph-0-0"> </g> <g id="UqoBzFCBTzJeN6VVm4-5IAae58Q=-glyph-0-1"> <path d="M 0.515625 1.515625 C 0.4375 1.875 0.375 1.96875 -0.109375 1.96875 C -0.25 1.96875 -0.375 1.96875 -0.375 2.203125 C -0.375 2.21875 -0.359375 2.3125 -0.234375 2.3125 C -0.078125 2.3125 0.09375 2.296875 0.25 2.296875 L 0.765625 2.296875 C 1.015625 2.296875 1.625 2.3125 1.875 2.3125 C 1.953125 2.3125 2.09375 2.3125 2.09375 2.109375 C 2.09375 1.96875 2.015625 1.96875 1.8125 1.96875 C 1.25 1.96875 1.21875 1.890625 1.21875 1.796875 C 1.21875 1.65625 1.75 -0.40625 1.828125 -0.6875 C 1.953125 -0.34375 2.28125 0.125 2.90625 0.125 C 4.25 0.125 5.71875 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4.921875 -1.4375 4.875 -1.4375 C 4.84375 -1.4375 4.828125 -1.421875 4.71875 -1.3125 C 3.953125 -0.296875 2.828125 -0.125 2.359375 -0.125 C 1.6875 -0.125 1.328125 -0.65625 1.328125 -1.546875 C 1.328125 -1.703125 1.328125 -2.015625 1.5 -2.765625 Z M 1.5625 -3.015625 C 2.078125 -4.859375 3.21875 -5.03125 3.59375 -5.03125 C 4.125 -5.03125 4.484375 -4.71875 4.484375 -4.265625 C 4.484375 -3.015625 2.5625 -3.015625 2.0625 -3.015625 Z M 1.5625 -3.015625 "></path> </g> <g id="UqoBzFCBTzJeN6VVm4-5IAae58Q=-glyph-1-0"> </g> <g id="UqoBzFCBTzJeN6VVm4-5IAae58Q=-glyph-1-1"> <path d="M 2.5 -5.078125 C 2.5 -5.296875 2.484375 -5.296875 2.265625 -5.296875 C 1.9375 -4.984375 1.515625 -4.796875 0.765625 -4.796875 L 0.765625 -4.53125 C 0.984375 -4.53125 1.40625 -4.53125 1.875 -4.734375 L 1.875 -0.65625 C 1.875 -0.359375 1.84375 -0.265625 1.09375 -0.265625 L 0.8125 -0.265625 L 0.8125 0 C 1.140625 -0.03125 1.828125 -0.03125 2.1875 -0.03125 C 2.546875 -0.03125 3.234375 -0.03125 3.5625 0 L 3.5625 -0.265625 L 3.28125 -0.265625 C 2.53125 -0.265625 2.5 -0.359375 2.5 -0.65625 Z M 2.5 -5.078125 "></path> </g> <g id="UqoBzFCBTzJeN6VVm4-5IAae58Q=-glyph-2-0"> </g> <g id="UqoBzFCBTzJeN6VVm4-5IAae58Q=-glyph-2-1"> <path d="M 7.875 -2.75 C 8.078125 -2.75 8.296875 -2.75 8.296875 -2.984375 C 8.296875 -3.234375 8.078125 -3.234375 7.875 -3.234375 L 1.40625 -3.234375 C 1.203125 -3.234375 0.984375 -3.234375 0.984375 -2.984375 C 0.984375 -2.75 1.203125 -2.75 1.40625 -2.75 Z M 7.875 -2.75 "></path> </g> </g> <clipPath id="UqoBzFCBTzJeN6VVm4-5IAae58Q=-clip-0"> <path clip-rule="nonzero" d="M 85 0 L 90.710938 0 L 90.710938 6 L 85 6 Z M 85 0 "></path> </clipPath> <clipPath id="UqoBzFCBTzJeN6VVm4-5IAae58Q=-clip-1"> <path clip-rule="nonzero" d="M 0 85 L 6 85 L 6 90.710938 L 0 90.710938 Z M 0 85 "></path> </clipPath> <clipPath id="UqoBzFCBTzJeN6VVm4-5IAae58Q=-clip-2"> <path clip-rule="nonzero" d="M 0 2 L 90.710938 2 L 90.710938 32 L 0 32 Z M 0 2 "></path> </clipPath> <clipPath 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87.875 C 5.671875 86.308594 4.402344 85.039062 2.835938 85.039062 C 1.269531 85.039062 0 86.308594 0 87.875 C 0 89.441406 1.269531 90.710938 2.835938 90.710938 C 4.402344 90.710938 5.671875 89.441406 5.671875 87.875 Z M 5.671875 87.875 "></path> </g> <g clip-path="url(#UqoBzFCBTzJeN6VVm4-5IAae58Q=-clip-2)"> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 88.197187 -5.465781 C 101.622969 -28.719687 153.497969 -28.719687 167.154219 -5.067344 " transform="matrix(1, 0, 0, -1, -82.205, 2.835)"></path> </g> <path fill="none" stroke-width="0.31879" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -1.194715 1.595019 C -1.094173 0.995697 -0.00167421 0.0988292 0.298787 0.00204245 C 0.000345966 -0.100805 -1.094793 -0.997643 -1.195316 -1.593649 " transform="matrix(-0.49998, -0.86601, -0.86601, 0.49998, 5.991, 8.3007)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#UqoBzFCBTzJeN6VVm4-5IAae58Q=-glyph-0-1" x="40.051" y="33.623"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#UqoBzFCBTzJeN6VVm4-5IAae58Q=-glyph-1-1" x="45.926" y="35.416"></use> </g> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 85.040937 -79.188437 L 85.040937 -6.309531 " transform="matrix(1, 0, 0, -1, -82.205, 2.835)"></path> <path fill="none" stroke-width="0.31879" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -1.19741 1.593233 C -1.095847 0.995576 0.00180875 0.101045 0.298684 -0.0005175 C 0.00180875 -0.0981737 -1.095847 -0.996611 -1.19741 -1.594267 " transform="matrix(0, -1, -1, 0, 2.83542, 9.14634)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#UqoBzFCBTzJeN6VVm4-5IAae58Q=-glyph-0-1" x="6.936" y="46.766"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#UqoBzFCBTzJeN6VVm4-5IAae58Q=-glyph-1-1" x="12.811" y="48.559"></use> </g> <g clip-path="url(#UqoBzFCBTzJeN6VVm4-5IAae58Q=-clip-3)"> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 90.693281 -83.52558 C 130.525312 -72.853705 157.8925 -45.482611 168.447187 -6.095892 " transform="matrix(1, 0, 0, -1, -82.205, 4.00567)"></path> </g> <path fill="none" stroke-width="0.31879" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -1.194787 1.593107 C -1.095705 0.996872 -0.00235559 0.100855 0.299712 -0.000196708 C -0.000861795 -0.100946 -1.09539 -0.99675 -1.194336 -1.593468 " transform="matrix(0.25879, -0.96593, -0.96593, -0.25879, 86.24256, 10.10195)"></path> <g fill="rgb(0%, 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0.996447 -0.000506736 0.101441 0.297788 -0.000621685 C 0.000987063 -0.10036 -1.097314 -0.997175 -1.196261 -1.593893 " transform="matrix(-0.25879, 0.96593, 0.96593, 0.25879, 4.46829, 80.60705)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#7GSLybxXgvUlUq-Y3WgEdkmwV00=-glyph-0-1" x="18.16" y="25.27233"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#7GSLybxXgvUlUq-Y3WgEdkmwV00=-glyph-1-1" x="24.035" y="27.06533"></use> </g> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 85.04 79.1865 L 85.04 6.3115 " transform="matrix(1, 0, 0, -1, 2.835, 87.874)"></path> <path fill="none" stroke-width="0.31879" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -1.195473 1.59333 C -1.097816 0.995674 -0.00016 0.101142 0.300621 -0.00042 C -0.00016 -0.0980763 -1.097816 -0.996514 -1.195473 -1.59417 " transform="matrix(0, 1, 1, 0, 87.87542, 81.56266)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#7GSLybxXgvUlUq-Y3WgEdkmwV00=-glyph-0-1" x="91.975" y="46.766"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#7GSLybxXgvUlUq-Y3WgEdkmwV00=-glyph-1-2" x="97.85" y="48.559"></use> </g> </svg> </div> <p>the third consisting of</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="106.484" height="90.709" viewBox="0 0 106.484 90.709"> <defs> <g> <g id="txoLAYpt7S6M7W7BR3XsdMxTfkg=-glyph-0-0"> </g> <g id="txoLAYpt7S6M7W7BR3XsdMxTfkg=-glyph-0-1"> <path d="M 0.515625 1.515625 C 0.4375 1.875 0.375 1.96875 -0.109375 1.96875 C -0.25 1.96875 -0.375 1.96875 -0.375 2.203125 C -0.375 2.21875 -0.359375 2.3125 -0.234375 2.3125 C -0.078125 2.3125 0.09375 2.296875 0.25 2.296875 L 0.765625 2.296875 C 1.015625 2.296875 1.625 2.3125 1.875 2.3125 C 1.953125 2.3125 2.09375 2.3125 2.09375 2.109375 C 2.09375 1.96875 2.015625 1.96875 1.8125 1.96875 C 1.25 1.96875 1.21875 1.890625 1.21875 1.796875 C 1.21875 1.65625 1.75 -0.40625 1.828125 -0.6875 C 1.953125 -0.34375 2.28125 0.125 2.90625 0.125 C 4.25 0.125 5.71875 -1.640625 5.71875 -3.390625 C 5.71875 -4.5 5.09375 -5.265625 4.203125 -5.265625 C 3.4375 -5.265625 2.78125 -4.53125 2.65625 -4.359375 C 2.5625 -4.953125 2.09375 -5.265625 1.609375 -5.265625 C 1.265625 -5.265625 0.984375 -5.109375 0.765625 -4.65625 C 0.546875 -4.21875 0.375 -3.484375 0.375 -3.4375 C 0.375 -3.390625 0.4375 -3.328125 0.515625 -3.328125 C 0.609375 -3.328125 0.625 -3.34375 0.6875 -3.625 C 0.875 -4.328125 1.09375 -5.03125 1.578125 -5.03125 C 1.859375 -5.03125 1.953125 -4.84375 1.953125 -4.484375 C 1.953125 -4.203125 1.90625 -4.078125 1.859375 -3.859375 Z M 2.578125 -3.734375 C 2.671875 -4.0625 3 -4.40625 3.1875 -4.578125 C 3.328125 -4.703125 3.71875 -5.03125 4.171875 -5.03125 C 4.703125 -5.03125 4.9375 -4.5 4.9375 -3.890625 C 4.9375 -3.3125 4.609375 -1.953125 4.296875 -1.34375 C 4 -0.6875 3.453125 -0.125 2.90625 -0.125 C 2.09375 -0.125 1.953125 -1.140625 1.953125 -1.1875 C 1.953125 -1.234375 1.984375 -1.328125 2 -1.390625 Z M 2.578125 -3.734375 "></path> </g> <g id="txoLAYpt7S6M7W7BR3XsdMxTfkg=-glyph-1-0"> </g> <g id="txoLAYpt7S6M7W7BR3XsdMxTfkg=-glyph-1-1"> <path d="M 2.25 -1.625 C 2.375 -1.75 2.703125 -2.015625 2.84375 -2.125 C 3.328125 -2.578125 3.796875 -3.015625 3.796875 -3.734375 C 3.796875 -4.6875 3 -5.296875 2.015625 -5.296875 C 1.046875 -5.296875 0.421875 -4.578125 0.421875 -3.859375 C 0.421875 -3.46875 0.734375 -3.421875 0.84375 -3.421875 C 1.015625 -3.421875 1.265625 -3.53125 1.265625 -3.84375 C 1.265625 -4.25 0.859375 -4.25 0.765625 -4.25 C 1 -4.84375 1.53125 -5.03125 1.921875 -5.03125 C 2.65625 -5.03125 3.046875 -4.40625 3.046875 -3.734375 C 3.046875 -2.90625 2.46875 -2.296875 1.515625 -1.34375 L 0.515625 -0.296875 C 0.421875 -0.21875 0.421875 -0.203125 0.421875 0 L 3.5625 0 L 3.796875 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clip-rule="nonzero" d="M 0 2 L 88 2 L 88 90.710938 L 0 90.710938 Z M 0 2 "></path> </clipPath> </defs> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" d="M 5.667969 2.835938 C 5.667969 1.269531 4.402344 0 2.835938 0 C 1.269531 0 0 1.269531 0 2.835938 C 0 4.402344 1.269531 5.667969 2.835938 5.667969 C 4.402344 5.667969 5.667969 4.402344 5.667969 2.835938 Z M 5.667969 2.835938 "></path> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" d="M 90.710938 2.835938 C 90.710938 1.269531 89.441406 0 87.875 0 C 86.308594 0 85.039062 1.269531 85.039062 2.835938 C 85.039062 4.402344 86.308594 5.667969 87.875 5.667969 C 89.441406 5.667969 90.710938 4.402344 90.710938 2.835938 Z M 90.710938 2.835938 "></path> <g clip-path="url(#txoLAYpt7S6M7W7BR3XsdMxTfkg=-clip-0)"> <path fill-rule="nonzero" fill="rgb(0%, 0%, 0%)" fill-opacity="1" d="M 90.710938 87.875 C 90.710938 86.308594 89.441406 85.039062 87.875 85.039062 C 86.308594 85.039062 85.039062 86.308594 85.039062 87.875 C 85.039062 89.441406 86.308594 90.710938 87.875 90.710938 C 89.441406 90.710938 90.710938 89.441406 90.710938 87.875 Z M 90.710938 87.875 "></path> </g> <g clip-path="url(#txoLAYpt7S6M7W7BR3XsdMxTfkg=-clip-1)"> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 79.387656 -83.527076 C 39.555625 -72.851295 12.188438 -45.484107 1.63375 -6.097389 " transform="matrix(1, 0, 0, -1, 2.835, 1.66433)"></path> </g> <path fill="none" stroke-width="0.31879" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -1.195804 1.593771 C -1.096858 0.997053 0.00144347 0.100238 0.298244 0.000499406 C -0.0000503311 -0.101563 -1.097173 -0.996569 -1.196254 -1.592804 " transform="matrix(-0.25879, -0.96593, -0.96593, 0.25879, 4.46829, 7.76061)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#txoLAYpt7S6M7W7BR3XsdMxTfkg=-glyph-0-1" x="18.16" y="65.91833"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#txoLAYpt7S6M7W7BR3XsdMxTfkg=-glyph-1-1" x="24.035" y="67.71133"></use> </g> <path fill="none" stroke-width="0.3985" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 85.04 -79.188437 L 85.04 -6.309531 " transform="matrix(1, 0, 0, -1, 2.835, 2.835)"></path> <path fill="none" stroke-width="0.31879" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -1.19741 1.59417 C -1.095847 0.996514 0.00180875 0.0980763 0.298684 0.00042 C 0.00180875 -0.101142 -1.095847 -0.995674 -1.19741 -1.59333 " transform="matrix(0, -1, -1, 0, 87.87542, 9.14634)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#txoLAYpt7S6M7W7BR3XsdMxTfkg=-glyph-0-1" x="91.975" y="46.766"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#txoLAYpt7S6M7W7BR3XsdMxTfkg=-glyph-1-2" x="97.85" y="48.559"></use> </g> </svg> </div> <p>and the fourth consisting of the following.</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="90.709" height="74.029" viewBox="0 0 90.709 74.029"> <defs> <g> <g id="VQ3SMExXn_SpZI4uBXpdD_3AwOs=-glyph-0-0"> </g> <g id="VQ3SMExXn_SpZI4uBXpdD_3AwOs=-glyph-0-1"> <path d="M 0.515625 1.515625 C 0.4375 1.875 0.375 1.96875 -0.109375 1.96875 C -0.25 1.96875 -0.375 1.96875 -0.375 2.203125 C -0.375 2.21875 -0.359375 2.3125 -0.234375 2.3125 C -0.078125 2.3125 0.09375 2.296875 0.25 2.296875 L 0.765625 2.296875 C 1.015625 2.296875 1.625 2.3125 1.875 2.3125 C 1.953125 2.3125 2.09375 2.3125 2.09375 2.109375 C 2.09375 1.96875 2.015625 1.96875 1.8125 1.96875 C 1.25 1.96875 1.21875 1.890625 1.21875 1.796875 C 1.21875 1.65625 1.75 -0.40625 1.828125 -0.6875 C 1.953125 -0.34375 2.28125 0.125 2.90625 0.125 C 4.25 0.125 5.71875 -1.640625 5.71875 -3.390625 C 5.71875 -4.5 5.09375 -5.265625 4.203125 -5.265625 C 3.4375 -5.265625 2.78125 -4.53125 2.65625 -4.359375 C 2.5625 -4.953125 2.09375 -5.265625 1.609375 -5.265625 C 1.265625 -5.265625 0.984375 -5.109375 0.765625 -4.65625 C 0.546875 -4.21875 0.375 -3.484375 0.375 -3.4375 C 0.375 -3.390625 0.4375 -3.328125 0.515625 -3.328125 C 0.609375 -3.328125 0.625 -3.34375 0.6875 -3.625 C 0.875 -4.328125 1.09375 -5.03125 1.578125 -5.03125 C 1.859375 -5.03125 1.953125 -4.84375 1.953125 -4.484375 C 1.953125 -4.203125 1.90625 -4.078125 1.859375 -3.859375 Z M 2.578125 -3.734375 C 2.671875 -4.0625 3 -4.40625 3.1875 -4.578125 C 3.328125 -4.703125 3.71875 -5.03125 4.171875 -5.03125 C 4.703125 -5.03125 4.9375 -4.5 4.9375 -3.890625 C 4.9375 -3.3125 4.609375 -1.953125 4.296875 -1.34375 C 4 -0.6875 3.453125 -0.125 2.90625 -0.125 C 2.09375 -0.125 1.953125 -1.140625 1.953125 -1.1875 C 1.953125 -1.234375 1.984375 -1.328125 2 -1.390625 Z M 2.578125 -3.734375 "></path> 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stroke-miterlimit="10" d="M 88.197187 5.46692 C 101.622969 28.720826 153.497969 28.720826 167.154219 5.068483 " transform="matrix(1, 0, 0, -1, -82.205, 38.18567)"></path> </g> <path fill="none" stroke-width="0.31879" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -1.196302 1.59308 C -1.095779 0.997074 -0.000640249 0.100235 0.301184 -0.00065869 C 0.000722588 -0.0974454 -1.09516 -0.996266 -1.195701 -1.595588 " transform="matrix(-0.49998, 0.86601, 0.86601, 0.49998, 5.991, 32.71997)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#VQ3SMExXn_SpZI4uBXpdD_3AwOs=-glyph-0-1" x="40.051" y="10.21967"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#VQ3SMExXn_SpZI4uBXpdD_3AwOs=-glyph-1-2" x="45.926" y="12.01367"></use> </g> </svg> </div> <p>The set of co-cones can be taken to be empty.</p> <p>In a model of this sketch, the leftmost vertex is sent to a one element set <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>, the middle vertex is sent to an arbitrary 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>, the top vertex is sent to the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">1 \times X</annotation></semantics></math>, the bottom vertex is sent to the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">X \times 1</annotation></semantics></math>, and the right vertex is sent to the product <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 \times X</annotation></semantics></math>. The arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> picks out an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">e_{X}</annotation></semantics></math> 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>.</p> <p>The arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow><annotation encoding="application/x-tex">e,-</annotation></semantics></math> is sent to an arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>X</mi></msub><mo>×</mo><mi>id</mi><mo>:</mo><mn>1</mn><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">e_{X} \times id: 1 \times X \rightarrow X \times X</annotation></semantics></math>, which is forced by the universal property of <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 \times X</annotation></semantics></math> and the fact that the diagrams</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="165.728" height="92.559" viewBox="0 0 165.728 92.559"> <defs> <g> <g id="dWL0wyoEs0hkvLStpYIqzRNpJQI=-glyph-0-0"> </g> <g id="dWL0wyoEs0hkvLStpYIqzRNpJQI=-glyph-0-1"> <path d="M 4.296875 -9.578125 C 4.296875 -9.921875 4.296875 -9.9375 4 -9.9375 C 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Similarly, the arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">-,e</annotation></semantics></math> is sent to an arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo>×</mo><msub><mi>e</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>×</mo><mn>1</mn><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">id \times e_{X}: X \times 1 \rightarrow X \times X</annotation></semantics></math>, which is forced by the fact that the diagrams</p> <div style="text-align: center"> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="165.728" height="89.068" viewBox="0 0 165.728 89.068"> <defs> <g> <g id="C5wMajoAzLm6AIEgcC5xIXBXZio=-glyph-0-0"> </g> <g 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class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a> to) powers 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">C = \emptyset</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is the set of all product cones.</p> </div> </p> <h2 id="properties">Properties</h2> <h3 id="RelationToLocallyRepresentableCategories">Relation to accessible and locally representable categories</h3> <p> <div class='num_prop' id='LimitSketchableCategoriesAreAccessible'> <h6>Proposition</h6> <p>The categories of models of sketches are equivalently the <a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a>.</p> </div> </p> <p> <div class='num_prop' id='LimitSketchableCategoriesAreLocallyPresentable'> <h6>Proposition</h6> <p>The categories of models of limit-sketches are the <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a>.</p> </div> (<a href="#AdámekRosický49">Adámek & Rosický 1994, Cor. 1.52</a>)</p> <div class="num_prop"> <h6 id="remark">Remark</h6> <p>From the discussion there we have that</p> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/accessible+category">accessible category</a> is equivalently:</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of a <a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a> that’s closed under <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>-filtered colimits for some <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></li> <li>the category of models of a sketch</li> </ul> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a> is equivalently:</p> <ul> <li>a <em>reflective</em> full subcategory of a presheaf category that’s closed under <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>-filtered colimits for some <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></li> <li>the category of models of a <em>limit</em> sketch</li> <li>an accessible category with all small limits</li> <li>an accessible category with all small colimits</li> </ul> </li> </ul> <p>We can “break in half” the difference between the two and define</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/locally+multipresentable+category">locally multipresentable category</a> to be equivalently: <ul> <li>a <em>multireflective</em> full subcategory of a presheaf category that’s closed under <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>-filtered colimits for some <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></li> <li>the category of models of a <em>limit and coproduct</em> sketch</li> <li>an accessible category with all small <em>connected</em> limits</li> <li>an accessible category with all small multicolimits</li> </ul> </li> </ul> <p>and</p> <ul> <li>a <em>weakly locally presentable category</em> to be equivalently: <ul> <li>a <em>weakly reflective</em> full subcategory of a presheaf category that’s closed under <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>-filtered colimits for some <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></li> <li>the category of models of a <em>limit and epi</em> sketch</li> <li>an accessible category with all small products</li> <li>an accessible category with all small weak colimits</li> </ul> </li> </ul> </div> <h3 id="monoidal_structures_on_the_category_of_sketches">Monoidal Structures on the Category of Sketches</h3> <p>The category of sketches is well behaved: it is complete, cocomplete, cartesian closed and has a second symmetric monoidal closed structure.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The category of sketches is <a class="existingWikiWord" href="/nlab/show/topological+category">topological</a> over the category of directed <a href="https://ncatlab.org/nlab/show/pseudograph">pseudographs</a>.</p> </div> <p>The above proposition gives the category of sketches Cartesian products - however these are often not the sketches one would expect when thinking of the product of two theories. Instead consider the tensor product:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>,</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S,T</annotation></semantics></math> be sketches. We define the sketch <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊗</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S \otimes T</annotation></semantics></math> to be:</p> <p>The vertices of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊗</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S \otimes T</annotation></semantics></math> are the product of the set of vertices from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, <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>. The set of arrows is given 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><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>α</mi><mo>∈</mo><mstyle mathvariant="sans-serif"><mi>Edge</mi></mstyle><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mi>b</mi><mo>∈</mo><mstyle mathvariant="sans-serif"><mi>Vertex</mi></mstyle><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>∪</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>a</mi><mo>∈</mo><mstyle mathvariant="sans-serif"><mi>Vertex</mi></mstyle><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mi>β</mi><mo>∈</mo><mstyle mathvariant="sans-serif"><mi>Edge</mi></mstyle><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \{ (\alpha, b) | \alpha \in \mathsf{Edge}(S), b \in \mathsf{Vertex}(T)\} \cup \{ (a, \beta) | a \in \mathsf{Vertex}(S), \beta \in \mathsf{Edge}(T)\} </annotation></semantics></math></div> <p>where the source of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\alpha,b)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s_S(\alpha), b)</annotation></semantics></math>, and vice versa. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is often called the <em>horizontal</em> structure and <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> as the <em>vertical</em> structure. The set of diagrams is the union of the following three sets:</p> <ul> <li> <p>The horizontal diagrams are constant in the second parameter: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>D</mi><mo>∈</mo><mstyle mathvariant="sans-serif"><mi>Diagrams</mi></mstyle><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mi>b</mi><mo>∈</mo><mstyle mathvariant="sans-serif"><mi>Vertex</mi></mstyle><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">H = \{ (D, b) | D \in \mathsf{Diagrams}(S), b \in \mathsf{Vertex}(T) \}</annotation></semantics></math></p> </li> <li> <p>The vertical diagrams are constant in the first parameter: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>a</mi><mo>∈</mo><mstyle mathvariant="sans-serif"><mi>Vertex</mi></mstyle><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mi>D</mi><mo>∈</mo><mstyle mathvariant="sans-serif"><mi>Diagrams</mi></mstyle><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V = \{ (a, D) | a \in \mathsf{Vertex}(S), D \in \mathsf{Diagrams}(T) \}</annotation></semantics></math></p> </li> <li> <p>Also add every square diagram: <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 set of squares 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xlink:href="#YwN5Xzuz6xNEYRreH_ZKs0q26K8=-glyph-4-3" x="77.221716" y="78.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#YwN5Xzuz6xNEYRreH_ZKs0q26K8=-glyph-5-1" x="81.75075" y="75.19675"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#YwN5Xzuz6xNEYRreH_ZKs0q26K8=-glyph-3-2" x="85.107" y="78.713"></use> </g> </svg> </div> <ul> <li>The set of cones and cocones are define analogously to the set of commuting diagrams, except <em>only</em> the vertical and horizontal cones are taken.</li> </ul> <p>This tensor product, along with the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>∅</mi><mo>,</mo><mi>∅</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\ast, \emptyset, \emptyset, \emptyset)</annotation></semantics></math>, gives the category of sketches a monoidal structure.</p> </div> <p>This monoidal structure is useful for considering structures like double categories (i.e. categories in the category of categories).</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>,</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S,T</annotation></semantics></math> be sketches, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> some category. Then the category of models of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> in the category of models of <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> 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> is equivalent to the category of models of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊗</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S \otimes T</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>One can ask when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊗</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">S \otimes T</annotation></semantics></math> has the same models as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>⊗</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">T \otimes S</annotation></semantics></math>, i.e. when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-models in the category of <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>-models are the same as <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>-models in the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-models. This is the case, roughly, when the colimits and limits specified in the sketches commute. For example, since limits always commute, you can swap the sketches if both are limit sketches. And you can swap a finite product sketch with a sifted colimit sketch etc. For more precise statements see David Bensons article and the references therein.</p> <p>In general one can not swap the order in the monoidal product. For example take the (terminal object + coproduct)-sketch <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> whose models are maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">X \to X \coprod 1</annotation></semantics></math> and the finite product sketch <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> whose models are monoids. Look at models of these 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>: Since in the category of monoids the terminal object is also initial, the coproduct of a monoid with the terminal object is isomorphic to that monoid again. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-models in <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>-Mod are monoids with an endomorphism. On the other hand <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>-models in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-Mod are pairs of monoids one of which has one element more, plus a homomorphism between them. These categories do not seem to be equivalent.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/internalization">internalization</a></li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Original articles:</p> <ul> <li id="Ehresmann68"> <p><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>, <em>Esquisses et types de structures algébriques</em>, Bul. Inst. Polit. Iasi <strong>14</strong> 1–2 (1968) 1-14 [<a class="existingWikiWord" href="/nlab/files/Ehresmann-EsquisseEtTypes.pdf" title="pdf">pdf</a>]</p> </li> <li id="BastianiEhresmann72"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9e+Bastiani">Andrée Bastiani</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>, <em>Categories of sketched structures</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome <strong>13</strong> no 2 (1972) 104-214 (<a href="http://www.numdam.org/item/?id=CTGDC_1972__13_2_104_0">numdam:CTGDC_1972__13_2_104_0</a>)</p> </li> </ul> <p>Overview in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Charles+Wells">Charles Wells</a>, <em>Sketches: Outline with references</em>, 1993 (<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.217.3531">citeseer:10.1.1.217.3531</a>, <a class="existingWikiWord" href="/nlab/files/Wells_Sketches.pdf" title="pdf">pdf</a>)</li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <em>Notes on Sketches</em>, Technical report of the Electrotechnical Laboratory (computer language section), Tsukuba, Japan (1997) [<a href="https://www.math.mcgill.ca/barr/papers/sketch.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Barr-NotesOnSketches.pdf" title="pdf">pdf</a>]</li> </ul> <p>Textbook accounts:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Wells">Charles Wells</a>, Section 4 of: <em><a class="existingWikiWord" href="/nlab/show/Toposes%2C+Triples%2C+and+Theories">Toposes, Triples, and Theories</a></em>, Originally published by: Springer-Verlag, New York, 1985, republished in: <p>Reprints in <a href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html">Theory and Applications of Categories, No. 12 (2005) pp. 1-287</a></p> </li> </ul> <p>and with emphasis on the relation to <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable</a> and <a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a>:</p> <ul> <li id="MakkaiPare"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Makkai">Michael Makkai</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Par%C3%A9">Robert Paré</a>, Chapter 3 onwards in: <em>Accessible categories: The foundations of categorical model theory</em>, Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989 (<a href="https://bookstore.ams.org/conm-104">ISBN:978-0-8218-7692-3</a>)</p> </li> <li id="AdámekRosický49"> <p><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Ad%C3%A1mek">Jiří Adámek</a>, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, Def. 1.49 onwards in: <em><a class="existingWikiWord" href="/nlab/show/Locally+presentable+and+accessible+categories">Locally presentable and accessible categories</a></em>, Cambridge University Press (1994)</p> </li> </ul> <p>Proof that not only every sketchable category is <a class="existingWikiWord" href="/nlab/show/accessible+category">accessible</a> but that conversely every <a class="existingWikiWord" href="/nlab/show/accessible+category">accessible category</a> is sketchable:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Christian+Lair">Christian Lair</a>, <em>Catégories modelables et catégories esquissables</em>, Diagrammes <strong>6</strong> 5 (1981) [<a href="http://www.numdam.org/item/?id=DIA_1981__6__A5_0">numdam:DIA_1981__6__A5_0</a>, <a href="http://www.numdam.org/article/DIA_1981__6__A5_0.pdf">pdf</a>]</li> </ul> <p>The tensor product of sketches is investigated here:</p> <ul> <li>David D. Benson, Multilinearity of sketches, <a href="http://www.tac.mta.ca/tac/volumes/1997/n11/n11.pdf">TAC, vol.11, nr. 3, pp.269-277</a></li> </ul> <p>The category of sketches itself was studied as a <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> for <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> in:</p> <ul> <li id="Gray"><a class="existingWikiWord" href="/nlab/show/John+W.+Gray">John W. Gray</a>, <em>The Category of Sketches as a Model for Algebraic Semantics</em>, Categories in Computer Science and Logic: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference Held June 14-20, 1987 with Support from the National Science Foundation. Vol. 92. American Mathematical Soc., 1989.</li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Martin+Brandenburg">Martin Brandenburg</a>, <em>Large limit sketches and topological space objects</em> (<a href="https://arxiv.org/abs/2106.11115">arXiv:2106.11115</a>)</li> </ul> <h3 id="in_enriched_category_theory">In enriched category theory</h3> <p>Discussion of sketches the generality of <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>:</p> <ul> <li id="Kelly82"> <p><a class="existingWikiWord" href="/nlab/show/G.+Max+Kelly">G. Max Kelly</a>, <em>Structures defined by finite limits in the enriched context, I</em>, Cahiers de topologie et géométrie différentielle <strong>23</strong> 1 (1982) 3-42 [<a href="http://archive.numdam.org/item/CTGDC_1982__23_1_3_0">numdam:CTGDC_1982__23_1_3_0</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <a class="existingWikiWord" href="/nlab/show/Carmen+Quinteriro">Carmen Quinteriro</a>, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, <em>A theory of enriched sketches</em>, Theory and Applications of Categories, <strong>4</strong> 3 (1998) 47-72 [<a href="http://www.tac.mta.ca/tac/volumes/1998/n3/4-03abs.html">tac:4-03</a>, <a href="http://www.tac.mta.ca/tac/volumes/1998/n3/n3.pdf">pdf</a>]</p> </li> </ul> <h3 id="generalisations">Generalisations</h3> <p>Discussion in the generality of <a class="existingWikiWord" href="/nlab/show/internalization">internalization</a> into <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a>:</p> <ul> <li id="Street76"> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Complete objects relative to a theory</em> (1976) [<a href="http://science.mq.edu.au/~street/Sketch.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Street-CompleteObjects.pdf" title="pdf">pdf</a>]</p> </li> <li id="KinoshitaPowerTakeyama99"> <p><a class="existingWikiWord" href="/nlab/show/Yoshiki+Kinoshita">Yoshiki Kinoshita</a>, <a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, Makoto Takeyama, <em>Sketches</em>, Journal of Pure and Applied Algebra <strong>143</strong> 1-3 (1999) 275-291 [<a href="https://doi.org/10.1016/S0022-4049(98)00114-5">doi:10.1016/S0022-4049(98)00114-5</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Pointwise extensions and sketches in bicategories</em> [<a href="https://arxiv.org/abs/1409.6427">arXiv:1409.6427</a>]</p> </li> </ul> <p>Generalising the approach of <a href="#KinoshitaPowerTakeyama99">Kinoshita, Power & Takeyama 1999</a>, a notion of sketch relative to an <a class="existingWikiWord" href="/nlab/show/algebraic+weak+factorisation+system">algebraic weak factorisation system</a> is defined in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Bourke">John Bourke</a>, <a class="existingWikiWord" href="/nlab/show/Richard+Garner">Richard Garner</a>, <em>Algebraic weak factorisation systems II: Categories of weak maps</em>, Journal of Pure and Applied Algebra <strong>220</strong> (2016) [<a href="http://arxiv.org/abs/1412.6560">arXiv:1412.6560</a>, <a href="https://doi.org/10.1016/j.jpaa.2015.06.003">doi:10.1016/j.jpaa.2015.06.003</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 2, 2024 at 06:11:41. 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