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href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a></p> </li> </ul> <h2 id='sidebar_universal_constructions'>Universal constructions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal construction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>/<a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a>/<a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+lifting+theorem'>adjoint lifting theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gabriel%E2%80%93Ulmer+duality'>Gabriel-Ulmer duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freyd-Mitchell+embedding+theorem'>Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> </ul> <h2 id='sidebar_applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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> </div> <h1 id='adhesive_categories'>Adhesive categories</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a></li><li><a href='#Examples'>Examples</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>An <em>adhesive category</em> is a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> in which <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushouts</a> of <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphisms</a> exist and “behave more or less as they do in the category of <a class='existingWikiWord' href='/nlab/show/diff/Set'>sets</a>”, or equivalently in any <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a>.</p> <h2 id='definition'>Definition</h2> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>The following conditions on a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> are equivalent. When they are satisfied, we say that <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is <strong>adhesive</strong>.</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullbacks</a> and <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushouts</a> of <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphisms</a>, and pushout squares of monomorphisms are also pullback squares and are stable under pullback.</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has pullbacks, and pushouts of monomorphisms, and the latter are also <a class='existingWikiWord' href='/nlab/show/diff/2-limit'>(bicategorical)</a> pushouts in the <a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a> of <a class='existingWikiWord' href='/nlab/show/diff/span'>spans</a> in <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> </li> <li> <p>(If <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is small) <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has pullbacks and pushouts of monomorphisms, and admits a <a class='existingWikiWord' href='/nlab/show/diff/full+and+faithful+functor'>full embedding</a> into a <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>Grothendieck topos</a> preserving pullbacks and pushouts of monomorphisms.</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has pullbacks and pushouts of monomorphisms, and in any cubical diagram:</p> <p><img alt='cube' src='https://i.imgur.com/doD9CSD.png' /></p> <p>if <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X\to Y</annotation></semantics></math> is a monomorphism, the bottom square is a pushout, and the left and back faces are pullbacks, then the top face is a pushout if and only if the front and right face are pullbacks. In other words, pushouts of monomorphisms are <a class='existingWikiWord' href='/nlab/show/diff/van+Kampen+colimit'>van Kampen colimits</a>.</p> </li> </ol> </div> <h2 id='properties'>Properties</h2> <p>Notice that generally <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphisms</a> are preserved by <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> (see <a href='monomorphism#MonomorphismsArePreservedByPullback'>there</a>) but in a general category they may not need to be preserved by <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a>. In an adhesive category, however, they are:</p> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p>In an adhesive category, suppose given a <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> square</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd><mover><mo>⟶</mo><mi>m</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mrow><msup><mrow /> <mi>f</mi></msup></mrow></mpadded><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><mpadded width='0'><mrow><msup><mrow /> <mi>g</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mi>n</mi></munder></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'><span> <del class='diffdel'> </del> \array{ C & \overset{m}{\longrightarrow} & A \\ \mathllap{{}^f}\big\downarrow && \big\downarrow\mathrlap{{}^g} \\ B & \underset{n}{\longrightarrow} & D }</span></annotation></semantics></math></div> <p>such that <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi></mrow><annotation encoding='application/x-tex'>m</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphism</a>. Then:</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> is also a monomorphism.</p> </li> <li> <p>The square is also a <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> square.</p> </li> <li> <p>The square is also a <a class='existingWikiWord' href='/nlab/show/diff/distributivity+pullback'>distributivity pullback</a> around <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>g</mi><mo>,</mo><mi>m</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(g,m)</annotation></semantics></math>; hence in particular <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><msub><mo>∀</mo> <mi>g</mi></msub><mi>m</mi></mrow><annotation encoding='application/x-tex'>n = \forall_g m</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/universal+quantifier'>universal quantification</a>.</p> </li> </ol> </div> <p>For a proof of the above proposition, see (<a href='#Lack'>Lack, prop. 2.1</a>) and (<a href='#LS1'>Lack-Sobocinski, Lemmas 2.3 and 2.8</a>). The latter Lemma 2.8 states only that <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><msub><mo>∀</mo> <mi>g</mi></msub><mi>m</mi></mrow><annotation encoding='application/x-tex'>n = \forall_g m</annotation></semantics></math> (a weaker universal property since it refers only to other <em>monomorphisms</em> into <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>), but the proof applies more generally.</p> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>An adhesive category with a <a class='existingWikiWord' href='/nlab/show/diff/strict+initial+object'>strict initial object</a> is automatically an <a class='existingWikiWord' href='/nlab/show/diff/extensive+category'>extensive category</a>.</p> </div> <p>We define a <a class='existingWikiWord' href='/nlab/show/diff/pushout+complement'>pushout complement</a> of <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>m:C\to A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>g:A\to D</annotation></semantics></math> to be a pair of arrows <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>f:C\to B</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>n:B\to D</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mi>f</mi><mo>=</mo><mi>g</mi><mi>m</mi></mrow><annotation encoding='application/x-tex'>n f = g m</annotation></semantics></math> and this <a class='existingWikiWord' href='/nlab/show/diff/commutative+square'>commutative square</a> is a <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a>. The following proposition is crucial in <a class='existingWikiWord' href='/nlab/show/diff/span+rewriting'>double pushout graph rewriting</a>.</p> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p>In an adhesive category, if <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>m:C\to A</annotation></semantics></math> is mono and <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>g:A\to D</annotation></semantics></math> is any morphism, then if a pushout complement exists, it is unique up to unique isomorphism.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>We give only a sketch; details are in <a href='#LS'>(LS, Lemma 4.5)</a>. If <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f,n)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>′</mo><mo>,</mo><mi>n</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f',n')</annotation></semantics></math> are two pushout complements, consider the two pushout squares as morphisms in the <a class='existingWikiWord' href='/nlab/show/diff/arrow+category'>arrow category</a> with target <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math>, and take their pullback. The resulting commutative cube can be viewed as a morphism in the category of commutative squares from the pullback square of <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi></mrow><annotation encoding='application/x-tex'>m</annotation></semantics></math> against itself (which is again <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi></mrow><annotation encoding='application/x-tex'>m</annotation></semantics></math>, since <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi></mrow><annotation encoding='application/x-tex'>m</annotation></semantics></math> is mono) to the pullback square of <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> against <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>n'</annotation></semantics></math>. Denote the vertex of the latter pullback square by <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>. Applying the van Kampen property in two directions, we find that the maps <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>U\to B</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>→</mo><mi>B</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>U\to B'</annotation></semantics></math> are both pushouts of <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>1_C</annotation></semantics></math>, hence isomorphisms. This gives an isomorphism between the pushout complements; it is unique since <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>n'</annotation></semantics></math> are mono (being pushouts of the mono <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi></mrow><annotation encoding='application/x-tex'>m</annotation></semantics></math>).</p> </div> <ins class='diffins'><div class='num_prop'> <h6 id='proposition_4'>Proposition</h6> <p>In an adhesive category, every monomorphism is <a class='existingWikiWord' href='/nlab/show/diff/regular+monomorphism'>regular</a>. In particular, every adhesive category is <a class='existingWikiWord' href='/nlab/show/diff/balanced+category'>balanced</a>.</p> </div></ins><ins class='diffins'> </ins><ins class='diffins'><div class='proof'> <h6 id='proof_2'>Proof</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_b11996978112ba0b52833f824f9082b984b8b504_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>m: A \to B</annotation></semantics></math> be a monomorphism. By adhesiveness, the cokernel pair (pushout)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b11996978112ba0b52833f824f9082b984b8b504_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>m</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mrow><msup><mrow /> <mi>m</mi></msup></mrow></mpadded><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><mpadded width='0'><mrow><msup><mrow /> <mi>u</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mi>v</mi></munder></mtd> <mtd><mi>P</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ A & \overset{m}{\longrightarrow} & B \\ \mathllap{{}^m}\big\downarrow && \big\downarrow\mathrlap{{}^u} \\ B & \underset{v}{\longrightarrow} & P \rlap{.} } </annotation></semantics></math></div> <p>is a pullback. This exhibits <math class='maruku-mathml' display='inline' id='mathml_b11996978112ba0b52833f824f9082b984b8b504_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi></mrow><annotation encoding='application/x-tex'>m</annotation></semantics></math> as an equalizer of <math class='maruku-mathml' display='inline' id='mathml_b11996978112ba0b52833f824f9082b984b8b504_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi></mrow><annotation encoding='application/x-tex'>u</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b11996978112ba0b52833f824f9082b984b8b504_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi></mrow><annotation encoding='application/x-tex'>v</annotation></semantics></math>.</p> </div></ins><ins class='diffins'> </ins><h2 id='Examples'>Examples</h2> <p>\begin{example} Any <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a> is adhesive (<a href='#LSToposes'>Lack-Sobocisnki</a>). For <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>Grothendieck toposes</a> this is easy, because <math class='maruku-mathml' display='inline' id='mathml_63cb033d95fc0f557b7b4ef346b03f74c2c8d40d_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math> is adhesive and adhesivity is a condition on colimits and finite limits, hence preserved by functor categories and left-exact localizations. For <a class='existingWikiWord' href='/nlab/show/diff/topos'>elementary toposes</a> it is a theorem of <a href='#LSToposes'>Lack and Sobocinski</a>.</p> <p>The fact that <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphisms</a> are preserved by <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushouts</a> in toposes plays a central role for <a class='existingWikiWord' href='/nlab/show/diff/Cisinski+model+structure'>Cisinski model structures</a> such as notably the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>classical model structure on simplicial sets</a>, where the <a class='existingWikiWord' href='/nlab/show/diff/class'>class</a> of monomorphisms is identified with the class of <a class='existingWikiWord' href='/nlab/show/diff/cofibration'>cofibrations</a> and as such required to be closed under pushout (in particular). \end{example}</p> <p>\begin{example} Some <a class='existingWikiWord' href='/nlab/show/diff/counterexample'>counter-examples</a>:</p> <ul> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/1-category'>1-category</a> <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a> of <a class='existingWikiWord' href='/nlab/show/diff/strict+category'>strict categories</a> and <a class='existingWikiWord' href='/nlab/show/diff/functor'>functors</a> is <em>not</em> adhesive.</p> </li> <li> <p>Neither are the categories of <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>posets</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>, and <a class='existingWikiWord' href='/nlab/show/diff/groupoid'>groupoids</a>.</p> </li> </ul> <p>\end{example} (<a href='#LS'>Lack and Sobocinski, Counterexample 7</a>).</p> <h2 id='related_concepts'>Related concepts</h2> <p>Adhesiveness is an <a class='existingWikiWord' href='/nlab/show/diff/exactness+property'>exactness property</a>, similar to being a <a class='existingWikiWord' href='/nlab/show/diff/regular+category'>regular category</a>, an <a class='existingWikiWord' href='/nlab/show/diff/exact+category'>exact category</a>, or an <a class='existingWikiWord' href='/nlab/show/diff/extensive+category'>extensive category</a>. In particular, it can be phrased in the language of “lex colimits”.</p> <h2 id='references'>References</h2> <ul> <li id='LS1'> <p><a class='existingWikiWord' href='/nlab/show/diff/Stephen+Lack'>Steve Lack</a> and <a class='existingWikiWord' href='/nlab/show/diff/Pawe%C5%82+Soboci%C5%84ski'>Pawel Sobocinski</a>, <em>Adhesive and quasiadhesive categories</em>, RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 3, pp. 511-545, <a href='http://www.numdam.org/item/ITA_2005__39_3_511_0/'>Numdam</a>, <a href='https://www.ioc.ee/~pawel/papers/adhesivejournal.pdf'>author PDF</a></p> </li> <li id='LS'> <p><a class='existingWikiWord' href='/nlab/show/diff/Stephen+Lack'>Steve Lack</a> and <a class='existingWikiWord' href='/nlab/show/diff/Pawe%C5%82+Soboci%C5%84ski'>Pawel Sobocinski</a>, <em>Adhesive categories</em>, In: Walukiewicz I. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2004. Lecture Notes in Computer Science, vol 2987. Springer, Berlin, Heidelberg. doi:<a href='https://doi.org/10.1007/978-3-540-24727-2_20'>10.1007/978-3-540-24727-2_20</a>, <a href='https://www.ioc.ee/~pawel/papers/adhesive.pdf'>author PDF</a></p> </li> <li id='LSToposes'> <p><a class='existingWikiWord' href='/nlab/show/diff/Stephen+Lack'>Steve Lack</a> and <a class='existingWikiWord' href='/nlab/show/diff/Pawe%C5%82+Soboci%C5%84ski'>Pawel Sobocinski</a>, <em>Toposes are adhesive</em>, In: Corradini A., Ehrig H., Montanari U., Ribeiro L., Rozenberg G. (eds) Graph Transformations. ICGT 2006. Lecture Notes in Computer Science, vol 4178. Springer, Berlin, Heidelberg. doi:<a href='https://doi.org/10.1007/11841883_14'>10.1007/11841883_14</a>, <a href='https://www.ioc.ee/~pawel/papers/toposesAdhesive.pdf'>author PDF</a></p> </li> <li id='Lack'> <p><a class='existingWikiWord' href='/nlab/show/diff/Stephen+Lack'>Steve Lack</a>, <em>An embedding theorem for adhesive categories</em>, Theory and Applications of Categories, Vol. 25, 2011, No. 7, pp 180-188. <a href='http://www.tac.mta.ca/tac/volumes/25/7/25-07abs.html'>journal page</a>, arXiv:<a href='https://arxiv.org/abs/1103.0600'>1103.0600</a></p> </li> <li id='GarnerLack12'> <p><a class='existingWikiWord' href='/nlab/show/diff/Richard+Garner'>Richard Garner</a>, <a class='existingWikiWord' href='/nlab/show/diff/Stephen+Lack'>Steve Lack</a>: <em>On the axioms for adhesive and quasiadhesive categories</em>, Theory and Applications of Categories, <strong>27</strong> 3 (2012) 27-46 [[arXiv:1108.2934](http://arxiv.org/abs/1108.2934), <a href='http://www.tac.mta.ca/tac/volumes/27/3/27-03abs.html'>tac:27-03</a>]</p> </li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on October 19, 2023 at 15:02:39. 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