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class="existingWikiWord" href="/nlab/show/allegory">allegory</a></p> <h2 id="types_of_binary_relation">Types of Binary relation</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflexive+relation">reflexive</a>, <a class="existingWikiWord" href="/nlab/show/irreflexive+relation">irreflexive</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+relation">symmetric</a>, <a class="existingWikiWord" href="/nlab/show/antisymmetric+relation">antisymmetric</a> <a class="existingWikiWord" href="/nlab/show/asymmetric+relation">asymmetric</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transitive+relation">transitive</a>, <a class="existingWikiWord" href="/nlab/show/comparison">comparison</a>;</p> </li> <li> <p>left and right <a class="existingWikiWord" href="/nlab/show/euclidean+relation">euclidean</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/total+relation">total</a>, <a class="existingWikiWord" href="/nlab/show/connected+relation">connected</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extensional+relation">extensional</a>, <a class="existingWikiWord" href="/nlab/show/well-founded+relation">well-founded</a> relations.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+relations">functional relations</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entire+relations">entire relations</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+relations">equivalence relations</a>, <a class="existingWikiWord" href="/nlab/show/congruence">congruence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/apartness+relations">apartness relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simple+graph">simple graph</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-congruence">2-congruence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-congruence">(n,r)-congruence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/relations+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="congruences">Congruences</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_pages'>Related pages</a></li> <li><a href='#see_also'>See also</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#in_common_mathematics'>In common mathematics</a></li> <li><a href='#in_category_theory'>In category theory</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, by <em>congruence</em> one means the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> on the collection of <a class="existingWikiWord" href="/nlab/show/subsets">subsets</a> of a <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> which regards two of these as equivalent if one is carried into the other by an <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a> (of the ambient Euclidean space).</p> <p>Similarly, in <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, by a <em>congruence</em> one means certain <a class="existingWikiWord" href="/nlab/show/equivalence+relations">equivalence relations</a> on elements of algebraic <a class="existingWikiWord" href="/nlab/show/structures">structures</a>, such as <a class="existingWikiWord" href="/nlab/show/groups">groups</a> or <a class="existingWikiWord" href="/nlab/show/rings">rings</a> (cf. e.g. the <a class="existingWikiWord" href="/nlab/show/multiplicative+group+of+integers+modulo+n">multiplicative group of integers modulo n</a>).</p> <p>Therefore, in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> the term <em>congruence</em> is used in the broad generality of <a class="existingWikiWord" href="/nlab/show/equivalence+relations">equivalence relations</a> on (the <a class="existingWikiWord" href="/nlab/show/generalized+elements">generalized elements</a> of) <em>any object</em> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <em>any</em> <a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finitely complete category</a>.</p> <h2 id="definitions">Definitions</h2> <div class="num_defn" id="Definition"> <h6 id="definition">Definition</h6> <p>In 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>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>, a <strong>congruence</strong> on an <a class="existingWikiWord" href="/nlab/show/object">object</a> <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 an <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> on <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> (i.e.: an <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> — hence an <a class="existingWikiWord" href="/nlab/show/internal+category">internal category</a> with all <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> being <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> — but with no non-<a class="existingWikiWord" href="/nlab/show/identity+morphism">identity</a> <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a>).</p> <p>This means that it consists of a <a class="existingWikiWord" href="/nlab/show/subobject">subobject</a></p> <div class="maruku-equation" id="eq:TheSubobject"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>R</mi><mover><mo>↪</mo><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> i \;\colon\; R\stackrel{(p_1,p_2)}\hookrightarrow X \times X </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> 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> with itself, equipped with the following <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>:</p> <ul> <li> <p>internal <a class="existingWikiWord" href="/nlab/show/reflexive+relation">reflexivity</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \colon X \to R</annotation></semantics></math> which is a <a class="existingWikiWord" href="/nlab/show/section">section</a> both of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math>, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mi>r</mi><mo>=</mo><msub><mi>p</mi> <mn>2</mn></msub><mi>r</mi><mo>=</mo><msub><mn>1</mn> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">p_1 r = p_2 r = 1_X</annotation></semantics></math>;</p> </li> <li> <p>internal <a class="existingWikiWord" href="/nlab/show/symmetric+relation">symmetry</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo lspace="verythinmathspace">:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">s \colon R \to R</annotation></semantics></math> which interchanges <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math>, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>∘</mo><mi>s</mi><mo>=</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_1\circ s = p_2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub><mo>∘</mo><mi>s</mi><mo>=</mo><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_2\circ s = p_1</annotation></semantics></math>;</p> </li> <li id="InternalTransitivity"> <p>internal <a class="existingWikiWord" href="/nlab/show/transitive+relation">transitivity</a>:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>R</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">t \,\colon\, R \times_X R \to R</annotation></semantics></math></p> <p>(where on the left we have the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>↪</mo><mi>X</mi><mo>×</mo><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">R \hookrightarrow X \times X \overset{p_2}{\to} X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>↪</mo><mi>X</mi><mo>×</mo><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">R \hookrightarrow X \times X \overset{p_1}{\to} X</annotation></semantics></math>, i.e. the subobject of pairs of <em>composable</em> pairs in relation)</p> <p>which factors the left/right projection map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>R</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">R \times_X R \to X \times X</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, i.e., the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>R</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>t</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mi>R</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>R</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><msub><mi>q</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><msub><mi>q</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && R \\ & {}^{\mathllap{t}}\nearrow & \big\downarrow \\ R \times_X R & \underset{(p_1 q_1,p_2 q_2)}{\longrightarrow} & X \times X \mathrlap{\,,} } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>q</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">q_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>q</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">q_2</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/projections">projections</a> defined by the <a class="existingWikiWord" href="/nlab/show/pullback"> pullback diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>R</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>R</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>q</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>R</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mrow></mrow> <mrow><msub><mi>q</mi> <mn>1</mn></msub></mrow></msup></mrow></mpadded></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mrow></mrow> <mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>R</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></munder></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ R \times_X R & \overset{q_2}\longrightarrow & R \\ \big\downarrow{\mathrlap{{}^{q_1}}} && \big\downarrow{\mathrlap{{}^{p_1}}} \\ R & \underset{p_2}\longrightarrow & X } </annotation></semantics></math></div></li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>, the maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>, <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>, 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> are necessarily unique if they exist.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Equivalently, a congruence on <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 an <a class="existingWikiWord" href="/nlab/show/internal+category">internal category</a> with <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 object of <a class="existingWikiWord" href="/nlab/show/objects">objects</a>, such that the (source,target)-map is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> and such that if there is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1 \to x_2</annotation></semantics></math> then there is also a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_2 \to x_1</annotation></semantics></math> (internally).</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>We can equivalently define a congruence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> as (a representing object of) a representable sub-presheaf of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hom(-, X \times X)</annotation></semantics></math> so that for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, the composite of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(Y) \hookrightarrow \hom(Y, X \times X) \cong \hom(Y, X) \times \hom(Y, X)</annotation></semantics></math> exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(Y)</annotation></semantics></math> as an equivalence relation on the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hom(Y, X)</annotation></semantics></math>. The upshot of this definition is that it makes sense even when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is not finitely complete.</p> </div> <div class="num_defn" id="EffectiveCongruence"> <h6 id="definition_2">Definition</h6> <p>A congruence which is the kernel pair of some morphism (example <a class="maruku-ref" href="#KernelPairIsCongruence"></a>) is called <strong>effective</strong>.</p> </div> <div class="num_defn" id="QuotientObject"> <h6 id="definition_3">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> of a congruence is called a <strong><a class="existingWikiWord" href="/nlab/show/quotient+object">quotient object</a></strong>.</p> <p>The quotient of an effective congruence is called an <strong>effective quotient</strong>.</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/regular+category">regular category</a> is called an <a class="existingWikiWord" href="/nlab/show/exact+category">exact category</a> if every congruence is effective.</p> </div> <h2 id="properties">Properties</h2> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>An effective congruence, def. <a class="maruku-ref" href="#EffectiveCongruence"></a>, is always the <a class="existingWikiWord" href="/nlab/show/kernel+pair">kernel pair</a> of its quotient, def. <a class="maruku-ref" href="#QuotientObject"></a>, if that quotient exists.</p> </div> <h2 id="examples">Examples</h2> <div class="num_example" id="DiagonalMorphismIsCongruence"> <h6 id="example">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/diagonal+morphism">diagonal morphism</a> on an object is a congruence and has a <a class="existingWikiWord" href="/nlab/show/quotient+object">quotient object</a> <a class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a> to the original object.</p> </div> <div class="num_example" id="KernelPairIsCongruence"> <h6 id="example_2">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/kernel+pair">kernel pair</a> is a congruence.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>An <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> is precisely a congruence in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> </div> <div class="num_example"> <h6 id="example_4">Example</h6> <p>The eponymous example is congruence modulo <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> (for a fixed <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>), which can be considered a congruence on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> in the category of <a class="existingWikiWord" href="/nlab/show/rigs">rigs</a>, or on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> in the category of <a class="existingWikiWord" href="/nlab/show/rings">rings</a>.</p> </div> <div class="num_example"> <h6 id="example_5">Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a> by a <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">K \hookrightarrow G</annotation></semantics></math> is the quotient of the congruence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>xy</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∈</mo><mi>K</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">R = \{(x,y) : xy^{-1} \in K \}</annotation></semantics></math>.</p> <p>Alternatively, a <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a> by a <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">K \hookrightarrow G</annotation></semantics></math> is the quotient of the congruence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>×</mo><mi>K</mi><mover><mo>↪</mo><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>G</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">G \times K \stackrel{(p_1,p_2)}{\hookrightarrow} G \times G</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math> is projection on the first factor and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math> is multiplication in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> (these are source and target maps in the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>⫽</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">G \sslash K</annotation></semantics></math>).</p> <p>A special case of this is that of a <em><a class="existingWikiWord" href="/nlab/show/quotient+module">quotient module</a></em>.</p> </div> <h2 id="related_pages">Related pages</h2> <ul> <li> <p>The notions of <a class="existingWikiWord" href="/nlab/show/regular+category">regular category</a> and <a class="existingWikiWord" href="/nlab/show/exact+category">exact category</a> can naturally be formulated in terms of congruences. A “higher arity” version, corresponding to <a class="existingWikiWord" href="/nlab/show/coherent+categories">coherent categories</a> and <a class="existingWikiWord" href="/nlab/show/pretoposes">pretoposes</a> is discussed at <a class="existingWikiWord" href="/nlab/show/familial+regularity+and+exactness">familial regularity and exactness</a>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/Mal%27cev+category">Mal'cev category</a> is a <a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finitely complete category</a> in which every internal relation satisfying reflexivity is thereby actually a congruence.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">Higher-categorical</a> generalizations are that of a <a class="existingWikiWord" href="/nlab/show/2-congruence">2-congruence</a> and of a <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-category</a>. See also <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-congruence">(n,r)-congruence</a>.</p> </li> </ul> <h2 id="see_also">See also</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/preordered+object">preordered object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+pseudo-equivalence+relation">internal pseudo-equivalence relation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/setoid+object">setoid object</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="in_common_mathematics">In common mathematics</h3> <ul> <li> <p>Wikipedia, <a href="https://en.wikipedia.org/wiki/Congruence_(geometry)">Congruence (geometry)</a></p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Congruence_relation">Congruence relation</a></em></p> </li> </ul> <h3 id="in_category_theory">In category theory</h3> <p>References using terminology as above:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jiri+Adamek">Jiri Adamek</a>, <a class="existingWikiWord" href="/nlab/show/Horst+Herrlich">Horst Herrlich</a>, <a class="existingWikiWord" href="/nlab/show/George+Strecker">George Strecker</a>, p. 195 of: <em><a class="existingWikiWord" href="/nlab/show/Abstract+and+Concrete+Categories">Abstract and Concrete Categories</a></em>, John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories <strong>17</strong> (2006) 1-507 [<a href="http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html">tac:tr17</a>, <a href="http://katmat.math.uni-bremen.de/acc/">book webpage</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Exp. 2.10.3.b in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em> Vol. 1 <em>Basic Category Theory</em>, Encyclopedia of Mathematics and its Applications <strong>50</strong>, Cambridge University Press (1994) [<a href="https://doi.org/10.1017/CBO9780511525858">doi:10.1017/CBO9780511525858</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Exps. 2.5.6 in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em> Vol. 2 <em>Categories and Structures</em>, Encyclopedia of Mathematics and its Applications <strong>50</strong>, Cambridge University Press (1994) [<a href="https://doi.org/10.1017/CBO9780511525865">doi:10.1017/CBO9780511525865</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marek+A.+Bednarczyk">Marek A. Bednarczyk</a>, <a class="existingWikiWord" href="/nlab/show/Andrzej+M.+Borzyszkowski">Andrzej M. Borzyszkowski</a>, <a class="existingWikiWord" href="/nlab/show/Wieslaw+Pawlowski">Wieslaw Pawlowski</a>, <em>Generalized congruences – epimorphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>at</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}at</annotation></semantics></math></em>, Theory and Applications of Categories <strong>5</strong> 11 (1999) 266-280 [<a href="http://www.tac.mta.ca/tac/volumes/1999/n11/5-11abs.html">tac:5-11</a>, <a href="https://eudml.org/doc/120226?lang=fr&limit=5">dml:120226</a>]</p> </li> </ul> <p>…</p> <p>But, for what it’s worth, a <em>different</em> use of the term “congruence” in category theory appears in Def. 3.5.1 on <a href="https://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf#page=107">p. 89</a> in:</p> <ul> <li id="BarrWells95"><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Wells">Charles Wells</a>, <em>Category theory for computing science</em>, Prentice-Hall International Series in Computer Science (1995); reprinted in: Reprints in Theory and Applications of Categories <strong>22</strong> (2012) 1-538 [<a href="http://www.math.mcgill.ca/barr/papers/ctcs.pdf">pdf</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html">tac:tr22</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 30, 2024 at 13:47:01. 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