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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="separators">Separators</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Terminology'>Caution on terminology</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#fibered'>In fibered categories</a></li> </ul> <li><a href='#examples_and_applications'>Examples and applications</a></li> <li><a href='#strengthened_separators'>Strengthened separators</a></li> <ul> <li><a href='#motivating_theorem'>Motivating theorem</a></li> </ul> <li><a href='#dense_separators'>Dense separators</a></li> <ul> <li><a href='#girauds_axioms'>Giraud’s axioms</a></li> </ul> <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 <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>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> (or family <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">\mathcal{S}</annotation></semantics></math> of objects) 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">\mathcal{C}</annotation></semantics></math> is called a <em>separator</em> or <em>generator</em> if <a class="existingWikiWord" href="/nlab/show/generalized+elements">generalized elements</a> with <a class="existingWikiWord" href="/nlab/show/domain">domain</a> <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> (or domain 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">\mathcal{S}</annotation></semantics></math>) are sufficient to distinguish <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/duality">dual</a> notion is that of a <em><a class="existingWikiWord" href="/nlab/show/coseparator">coseparator</a></em>.</p> <h2 id="Terminology">Caution on terminology</h2> <p>The term ‘generator’ is slightly more ambiguous because of the use of ‘generators’ in <a class="existingWikiWord" href="/nlab/show/generators+and+relations">generators and relations</a>. That said, there is a link between these two senses provided by theorem <a class="maruku-ref" href="#motive"></a> (q.v.).</p> <h2 id="definitions">Definitions</h2> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>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>S</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">S \in \mathcal{C}</annotation></semantics></math> of 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">\mathcal{C}</annotation></semantics></math> is called a <strong>separator</strong> or a <strong>generator</strong> or a <strong>separating object</strong> or a <strong>generating object</strong>, or is said to <strong>separate morphisms</strong> if:</p> <ul> <li>for every pair of <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g \colon X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>e</mi><mo>=</mo><mi>g</mi><mo>∘</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">f\circ e = g\circ e</annotation></semantics></math> for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">e\colon S \to X</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f = g</annotation></semantics></math></li> </ul> </div> <p>Equivalently, we have that <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 a separator if, for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f:S\rightarrow X</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/jointly+epimorphic+family">jointly epic</a>. Assuming that <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 class="existingWikiWord" href="/nlab/show/locally+small">locally small category</a>, we have equivalently that <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 a separator if the <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">Hom(S,-) \colon \mathcal{C} \to </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> is <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful</a>.</p> <p>More generally:</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/family">family</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒮</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>a</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mi>a</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{S} = (S_a)_{(a \in A)}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> of 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">\mathcal{C}</annotation></semantics></math> is a <strong>separating family</strong> or a <strong>generating family</strong> if:</p> <ul> <li>for every pair of <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g \colon X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>e</mi><mo>=</mo><mi>g</mi><mo>∘</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">f \circ e = g \circ e</annotation></semantics></math> for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo lspace="verythinmathspace">:</mo><msub><mi>S</mi> <mi>a</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">e \colon S_a \to X</annotation></semantics></math> sourced in the family, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f = g</annotation></semantics></math>.</li> </ul> </div> <p>Assuming again that <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 class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a>, we have equivalently that <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">\mathcal{S}</annotation></semantics></math> is a separating family if the family of <a class="existingWikiWord" href="/nlab/show/hom+functors">hom functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>a</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">Hom(S_a,-) \colon \mathcal{C} \to </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math>) is <a class="existingWikiWord" href="/nlab/show/jointly+faithful+family+of+functors">jointly faithful</a>.</p> <p>Since repetition is irrelevant in a separating family, we may also speak of a <em>separating <a class="existingWikiWord" href="/nlab/show/class">class</a></em> instead of a separating family.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A <strong>separating set</strong> is a <a class="existingWikiWord" href="/nlab/show/size+issues">small</a> separating class.</p> </div> <h3 id="fibered">In fibered categories</h3> <p>The notion of separating family can be generalized from categories to <a class="existingWikiWord" href="/nlab/show/fibered+categories">fibered categories</a> in such a way that the <a class="existingWikiWord" href="/nlab/show/family+fibration">family fibration</a> of a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> has a separating family if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> has a small separating family.</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>A separating family in a <a class="existingWikiWord" href="/nlab/show/fibered+category">fibered category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>:</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">P:\mathbf{E}\to \mathbf{B}</annotation></semantics></math> is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle></mrow><annotation encoding="application/x-tex">S\in \mathbf{E}</annotation></semantics></math> such that for every parallel pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f,g:A\to B</annotation></semantics></math> in <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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>≠</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f\neq g</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(f) = P(g)</annotation></semantics></math> there exist arrows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">c: X\to S</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">h:X\to A</annotation></semantics></math> (constituting a span) such that <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>-cartesian, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>h</mi><mo>≠</mo><mi>g</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">f h \neq g h</annotation></semantics></math> .</p> </div> <p>See Definition B2.4.1 in the <a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a>.</p> <h2 id="examples_and_applications">Examples and applications</h2> <ul> <li> <p>In <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, any <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited set</a> is a separator; in particular, the <a class="existingWikiWord" href="/nlab/show/point">point</a> is a separator.</p> </li> <li> <p>More generally, in any <a class="existingWikiWord" href="/nlab/show/well-pointed+category">well-pointed category</a>, any <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> is a separator. More generally still, in any represented <a class="existingWikiWord" href="/nlab/show/concrete+category">concrete category</a>, the representing object is a separator.</p> </li> <li> <p>The standard example of a separator in the category of <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>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> over a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <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> is any <a class="existingWikiWord" href="/nlab/show/free+module">free module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">R^I</annotation></semantics></math> (for <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> an <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited set</a>) and <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> (which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">R^I</annotation></semantics></math> for <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> a <a class="existingWikiWord" href="/nlab/show/point">point</a>) in particular. If a separator is a finitely generated <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a> in the category of <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>-modules, then one sometimes says (especially in the older literature, e.g. Freyd’s <em>Abelian Categories</em>) that the separator is a <em>progenerator</em>. Progenerators are important in classical Morita theory, see <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>.</p> </li> <li> <p>The existence of a small separating family is one of the conditions in <a class="existingWikiWord" href="/nlab/show/Giraud%27s+theorem">Giraud's theorem</a> characterizing <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a>es.</p> </li> <li> <p>The existence of a small (co)separating family is one of the conditions in one version of the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a>.</p> </li> <li> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Set^{op}</annotation></semantics></math>, the two-element set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math> is a separator, thus every <a class="existingWikiWord" href="/nlab/show/continuous+functor">continuous functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>Set</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F:Set\to C</annotation></semantics></math> into a locally small 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">C</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> by the special <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a>.</p> </li> <li> <p>On the other hand, the opposite of the category of groups does not admit a separator, since there exists a continuous functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ML</mi><mo>:</mo><mi>Group</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">ML: Group\to Set</annotation></semantics></math> which does not have a left adjoint (see Example 3.1 on the page <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a>).</p> </li> </ul> <h2 id="strengthened_separators">Strengthened separators</h2> <div class="num_theorem" id="motive"> <h6 id="motivating_theorem">Motivating theorem</h6> <p>If <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 <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a> and has all small <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>s, then a set-indexed family <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>a</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">(S_a)_{(a\colon A)}</annotation></semantics></math> is separating if and only if, for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X\in C</annotation></semantics></math>, the canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ε</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo lspace="verythinmathspace">:</mo><msub><mi>S</mi> <mi>a</mi></msub><mo>→</mo><mi>X</mi></mrow></munder><msub><mi>S</mi> <mi>a</mi></msub><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \varepsilon_X\colon \coprod_{a\colon A, f\colon S_a \to X} S_a \longrightarrow X </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>.</p> </div> <p>This theorem explains a likely origin of the term “generator” or “generating family”. For example, in linear algebra, one says that a set of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>a</mi></msub><mo>:</mo><msub><mi>S</mi> <mi>a</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f_a: S_a \to X</annotation></semantics></math> spans or generates <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> if the induced map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊕</mo><msub><mi>S</mi> <mi>a</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\oplus S_a \to X</annotation></semantics></math> maps epimorphically onto <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>More generally:</p> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>If <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">\mathcal{E}</annotation></semantics></math> is a subclass of epimorphisms, we say that <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>a</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">(S_a)_{(a\colon A)}</annotation></semantics></math> is an <em><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">\mathcal{E}</annotation></semantics></math>-separator</em> or <strong><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">\mathcal{E}</annotation></semantics></math>-generator</strong> if each morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ε</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\varepsilon_X</annotation></semantics></math> (as above) is in <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">\mathcal{E}</annotation></semantics></math>.</p> </div> <p>The weakest commonly-seen strengthened notion is that of <strong>extremal separator</strong>, i.e. separator where all maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ε</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\varepsilon_X</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/extremal+epimorphisms">extremal epimorphisms</a>. The notion of extremal separator admits an equivalent reformulation not referencing coproducts:</p> <div class="num_prop" id="Extremal"> <h6 id="proposition">Proposition</h6> <p>If <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 <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a> and has all small <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>s, then a set-indexed family <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>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">(S_i)_{(i\colon I)}</annotation></semantics></math> is an extremal separator if and only if the functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi mathvariant="normal">Set</mi></mrow><annotation encoding="application/x-tex">C(S_i,-):C\to\mathrm{Set}</annotation></semantics></math> are jointly faithful and jointly <a class="existingWikiWord" href="/nlab/show/conservative+functor">conservative</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Assume first that the family <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>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">(S_i)_{(i\colon I)}</annotation></semantics></math> is an extremal separator. The functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi mathvariant="normal">Set</mi></mrow><annotation encoding="application/x-tex">C(S_i,-):C\to\mathrm{Set}</annotation></semantics></math> are jointly faithful for every separator. To see that they are also jointly conservative, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A\to B</annotation></semantics></math> such that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(S_i,f)</annotation></semantics></math> are bijective. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ε</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\varepsilon_B</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> since all its components do, which implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is an extremal epi since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ε</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\varepsilon_B</annotation></semantics></math> is one by assumption. It remains to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a monomorphism. For this, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">u,v:X\to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>u</mi><mo>=</mo><mi>f</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">f u = f v</annotation></semantics></math>. Then we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>u</mi><mi>h</mi><mo>=</mo><mi>f</mi><mi>v</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">f u h = f v h</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i\in I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">h:S_i\to X</annotation></semantics></math>, which implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>h</mi><mo>=</mo><mi>v</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">u h = v h</annotation></semantics></math> since the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(S_i,f)</annotation></semantics></math> are bijective, and we conclude that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>=</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">u=v</annotation></semantics></math> since <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>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">(S_i)_{(i\colon I)}</annotation></semantics></math> is separating.</p> <p>Conversely, assume that the functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(S_i,-)</annotation></semantics></math> are jointly faithful and jointly conservative. Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A\in C</annotation></semantics></math>, joint faithfulness implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ε</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\varepsilon_A</annotation></semantics></math> is epic. To see that it is extremally so, assume a factorization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ε</mi> <mi>A</mi></msub><mo>=</mo><mi>m</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">\varepsilon_A = m g</annotation></semantics></math> with <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> monic. We have to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> is an isomorphism, and for this it is sufficient to show that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(S_i,m)</annotation></semantics></math> are bijections. Injectivity is clear since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> is monic, and surjectivity follows since every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">h:S_i\to A</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ε</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\varepsilon_A</annotation></semantics></math>.</p> </div> <p>The concepts “strong separator” and “regular separator” corresponding to the notions of <a class="existingWikiWord" href="/nlab/show/strong+epimorphism">strong epimorphism</a> and <a class="existingWikiWord" href="/nlab/show/regular+epimorphism">regular epimorphism</a> do not admit such a reformulation, but the following result shows that they are equivalent to extremal separators in reasonable categories.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Assume that <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 <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small</a> and has all small <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>s.</p> <ol> <li> <p>If <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 balanced, then every separator is extremal.</p> </li> <li> <p>If <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> has pullbacks, then every extremal separator is strong.</p> </li> <li> <p>If <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 <a class="existingWikiWord" href="/nlab/show/regular+category">regular</a>, then every strong separator is regular.</p> </li> </ol> <p>The converse implications do always hold.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is a direct consequence of the facts that</p> <ol> <li> <p>in a balanced category every epi is extremal,</p> </li> <li> <p>in a category with pullbacks, every extremal epi is strong, and</p> </li> <li> <p>in a regular category every strong epi is regular.</p> </li> </ol> </div> <div class="un_remark"> <h6 id="remarks">Remarks</h6> <ol> <li> <p>Proposition <a class="maruku-ref" href="#Extremal"></a> gives rise to a notion of extremal separator that makes sense independently of the existence of coproducts. In fact claim 1 of the preceding result holds in this more general setting, since every faithful functor out of a balanced category is conservative.</p> </li> <li> <p>Most of the literature uses the term “strong separator” (or strong generator) for what we call an extremal separator. <a href="#AdamekRosicky">Adamek and Rosicky (Section 0.6)</a> also comment on this mismatch, writing “It would be more reasonable, but unfortunately less standard, to call [a strong generator] an extremal generator”. However, item 2 of the preceding result shows that this discrepancy disappears and the terms coincide in presence of pullbacks (and coproducts).</p> </li> <li> <p>In the Elephant, Johnstone uses “separator” in the same sense as we do, and writes “generator” for extremal separators, in the more general sense not assuming coproducts. Since he always assumes finite limits, he can use a simplified criterion only requiring joint conservativity of the hom-functors (since a conservative functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F:C\to D</annotation></semantics></math> is automatically faithful whenever <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> has equalizers and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> preserves them).</p> </li> </ol> </div> <h3 id="dense_separators">Dense separators</h3> <p>Finally, the strongest kind of separator commonly seen is that of <strong>dense separator</strong>.</p> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p>A <strong>dense separator</strong> 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">C</annotation></semantics></math> is a family <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>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">(S_i)_{(i\colon I)}</annotation></semantics></math> of objects such that the generated full subcategory is <a class="existingWikiWord" href="/nlab/show/dense+subcategory">dense</a>.</p> </div> <p>Every dense separator is an extremal separator, and it is also strong and regular whenever those words make sense, i.e. <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 locally small and has small coproducts. If <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> furthermore has pullbacks and the coproducts are pullback-stable, then every regular separator is dense (see <a href="#Borceux">Borceux I, Proposition 4.5.6</a>). To see that the pullback-stability condition is necessary, consider the category of abelian groups. Here, the free group on one generator is a regular, but not a dense separator.</p> <h3 id="girauds_axioms">Giraud’s axioms</h3> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos#Giraud">Giraud's axioms</a> characterize Grothendieck toposes as locally small regular categories with effective equivalence relations and disjoint and pullback-stable coproducts admitting a small separator. The previously stated and cited results show that in fact every such separator is dense (the effectivity and disjointness assumptions don’t play a role for this conclusion).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/coseparator">coseparator</a></li> </ul> <h2 id="references">References</h2> <ul> <li id="Borceux"> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <em>Handbook of Categorical Algebra I</em>, Cambridge University Press, 1994</p> </li> <li id="AdamekRosicky"> <p><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Ad%C3%A1mek">Jiří Adámek</a> and <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosicky">Jiří Rosicky</a>, Locally presentable and accessible categories, Cambridge University Press</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 18, 2023 at 11:55:07. 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