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this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <h1 id="cores">Cores</h1> <p>The <strong>core</strong> of a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the maximal subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> that is a groupoid, consisting of all its objects and all isomorphisms between them. The core is not a functor on the 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>, but it is a functor on the (2,1)-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cat</mi> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex">Cat_g</annotation></semantics></math> of categories, functors, and natural isomorphisms. In fact, it is a coreflection of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cat</mi> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex">Cat_g</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gpd</mi></mrow><annotation encoding="application/x-tex">Gpd</annotation></semantics></math>.</p> <p>In general, for any 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex">K_g</annotation></semantics></math> for its “homwise core:” the sub-2-category determined by all the objects and morphisms but only the iso 2-cells. Of course, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex">K_g</annotation></semantics></math> is a (2,1)-category. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gpd</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">gpd(K)</annotation></semantics></math> is a full sub-2-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex">K_g</annotation></semantics></math>, and we can ask whether it is coreflective. In a <a class="existingWikiWord" href="/michaelshulman/show/regular+2-category">regular 2-category</a>, however, it turns out that there is a stronger and more useful notion which implies this coreflectivity.</p> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>A <strong>core</strong> of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> in a regular 2-category is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">J\to A</annotation></semantics></math> which is eso, <a class="existingWikiWord" href="/nlab/show/pseudomonic+functor">pseudomonic</a>, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is groupoidal.</p> </div> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>In a regular 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, any core <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">J\to A</annotation></semantics></math> is a coreflection of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex">K_g</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gpd</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">gpd(K)</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We must show that for any groupoidal <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>, the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>gpd</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>K</mi> <mi>g</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>K</mi> <mi>g</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">gpd(K)(G,J)=K_g(G,J)\to K_g(G,A)</annotation></semantics></math></div> <p>is an equivalence. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">J\to A</annotation></semantics></math> is pseudomonic in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, it is ff in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex">K_g</annotation></semantics></math>, so this functor is full and faithful; thus it remains to show it is eso. Given any map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">G\to A</annotation></semantics></math>, take the pullback</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>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>J</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{P & \to & J\\ \downarrow && \downarrow \\ G & \to & A}</annotation></semantics></math></div> <p>and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo>⇉</mo><mspace width="thickmathspace"></mspace><mi>P</mi></mrow><annotation encoding="application/x-tex">P_1\;\rightrightarrows\; P</annotation></semantics></math> be the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">P\to G</annotation></semantics></math>. Since the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">P\to A</annotation></semantics></math> descends to <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>, it comes equipped with an action by this kernel. However, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is groupoidal, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo>⇉</mo><mspace width="thickmathspace"></mspace><mi>P</mi></mrow><annotation encoding="application/x-tex">P_1\;\rightrightarrows\; P</annotation></semantics></math> is a <a class="existingWikiWord" href="/michaelshulman/show/n-congruence">(2,1)-congruence</a>, so the 2-cell defining the action must be an isomorphism. Therefore, it must factor uniquely through the pseudomonic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">J\to A</annotation></semantics></math>, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">P\to J</annotation></semantics></math> has an action as well; thus it descends to a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">G\to J</annotation></semantics></math>, as desired.</p> </div> <p>In particular, cores in a regular 2-category are unique up to unique equivalence. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J(A)</annotation></semantics></math> for the core of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, when it exists.</p> <div class="num_lemma"> <h6 id="lemma_2">Lemma</h6> <p>An object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of a (2,1)-exact 2-category has a core if and only if there is an eso <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C\to A</annotation></semantics></math> where <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 groupoidal.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>“Only if” is clear, so suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">p:C\to A</annotation></semantics></math> is eso and <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 groupoidal. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>=</mo><mi>C</mi><msub><mo>×</mo> <mi>A</mi></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">C_1 = C\times_A C</annotation></semantics></math> be the pullback. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math> is also groupoidal and is a (2,1)-congruence on <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>, so by exactness it is the kernel of some eso <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">q:C\to J</annotation></semantics></math>. There is an evident induced map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><mi>J</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">m:J\to A</annotation></semantics></math>; we claim that this is a core of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>Evidently <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><mi>J</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">m:J\to A</annotation></semantics></math> is eso, since the eso <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C\to A</annotation></semantics></math> factors through it. And since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math> is a (2,1)-congruence, the classification of <a class="existingWikiWord" href="/michaelshulman/show/n-congruence">congruences</a> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is groupoidal; thus it remains 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 pseudomonic.</p> <p>First suppose given <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>X</mi><mspace width="thickmathspace"></mspace><mo>⇉</mo><mspace width="thickmathspace"></mspace><mi>J</mi></mrow><annotation encoding="application/x-tex">f,g: X\;\rightrightarrows\; J</annotation></semantics></math>. Pulling back <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> gives esos <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>X</mi></mrow><annotation encoding="application/x-tex">P_1\to X</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>X</mi></mrow><annotation encoding="application/x-tex">P_2\to X</annotation></semantics></math>, whose pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>=</mo><msub><mi>P</mi> <mn>1</mn></msub><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>P</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">P = P_1\times_X P_2</annotation></semantics></math> comes with an eso <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>:</mo><mi>P</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">r:P \to T</annotation></semantics></math> and two morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>,</mo><mi>k</mi><mo>:</mo><mi>P</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">h,k:P \to C</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>h</mi><mo>≅</mo><mi>f</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">q h \cong f r</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>k</mi><mo>≅</mo><mi>g</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">q k \cong g r</annotation></semantics></math>. Then any pair of 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>:</mo><mi>f</mi><mo>→</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">\alpha,\beta: f\to g</annotation></semantics></math> induce maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mspace width="thickmathspace"></mspace><mo>⇉</mo><mspace width="thickmathspace"></mspace><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">P\;\rightrightarrows\; C_1</annotation></semantics></math>, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math> is the kernel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">/</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(q/q)</annotation></semantics></math>. But if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mi>α</mi><mo>=</mo><mi>m</mi><mi>β</mi></mrow><annotation encoding="application/x-tex">m\alpha = m\beta</annotation></semantics></math>, then these two maps must be equal, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math> is also the kernel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p/p)</annotation></semantics></math>. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mi>r</mi><mo>=</mo><mi>β</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">\alpha r=\beta r</annotation></semantics></math>, and since <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 eso, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>=</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\alpha=\beta</annotation></semantics></math>; thus <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 faithful.</p> <p>On the other hand, again given <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>X</mi><mspace width="thickmathspace"></mspace><mo>⇉</mo><mspace width="thickmathspace"></mspace><mi>J</mi></mrow><annotation encoding="application/x-tex">f,g: X\;\rightrightarrows\; J</annotation></semantics></math>, any isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>m</mi><mi>f</mi><mo>≅</mo><mi>m</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">\alpha: m f\cong m g</annotation></semantics></math> induces a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">P\to C_1</annotation></semantics></math> and therefore a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>:</mo><mi>f</mi><mi>r</mi><mo>→</mo><mi>g</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">\beta: f r\to g r</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mi>β</mi><mo>=</mo><mi>α</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">m\beta = \alpha r</annotation></semantics></math>. To show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> descends to a 2-cell <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\to g</annotation></semantics></math>, we must verify that it is an action 2-cell for the actions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mspace width="thickmathspace"></mspace><mo>⇉</mo><mspace width="thickmathspace"></mspace><mi>J</mi></mrow><annotation encoding="application/x-tex">P\;\rightrightarrows\; J</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">f r</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">g r</annotation></semantics></math>. But <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mi>β</mi></mrow><annotation encoding="application/x-tex">m\beta</annotation></semantics></math> is an action 2-cell, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mi>β</mi><mo>=</mo><mi>α</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">m\beta = \alpha r</annotation></semantics></math>, and we now know 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 faithful, so it reflects the axiom for an action 2-cell. Therefore, <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 full-on-isomorphisms, and hence pseudomonic.</p> </div> <h1 id="enough_groupoids">Enough groupoids</h1> <p>We say that a regular 2-category has <strong>enough groupoids</strong> if every object admits an eso from a groupoidal one. Thus, a (2,1)-exact 2-category has cores if and only if it has enough groupoids.</p> <p>Likewise, we say that a regular 2-category has <strong>enough discretes</strong> if every object admits an eso from a discrete one. Clearly this is a stronger condition. Note, though, that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> has enough groupoids, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pos</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">pos(K)</annotation></semantics></math> has enough discretes, since the core of any posetal object is discrete.</p> <p>Having enough discretes, or at least enough groupoids, is a very familiar aspect of 2-categories such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>. It also turns out to make the <a class="existingWikiWord" href="/michaelshulman/show/2-internal+logic">internal logic</a> noticeably easier to work with. However, in a sense none of the really “new” 2-categories have enough groupoids or discretes.</p> <ul> <li> <p>The 2-exact 2-categories having enough discretes are precisely the categories of internal categories and anafunctors in 1-exact 1-categories; see <a class="existingWikiWord" href="/michaelshulman/show/exact+completion+of+a+2-category">exact completion of a 2-category</a>. Likewise, any 2-exact 2-categories having enough groupoids consists of certain internal categories in a (2,1)-category.</p> </li> <li> <p>Basically the only <a class="existingWikiWord" href="/michaelshulman/show/Grothendieck+2-topos">Grothendieck 2-topos</a>es having enough discretes are the 2-categories of stacks on <a class="existingWikiWord" href="/michaelshulman/show/2-site">1-sites</a>, and the only ones having enough groupoids are the 2-categories of stacks on (2,1)-sites. The “honestly 2-dimensional” case of stacks on 2-sites (almost?) never have enough of either.</p> </li> </ul> <h1 id="subobjects">Subobjects</h1> <p>We also remark that cores, when they exist, “capture all the information about subobjects.”</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a regular 2-category and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an object having a core <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><mi>J</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">j:J\to A</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mo>*</mo></msup><mo>:</mo><mi>Sub</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sub</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j^*:Sub(A)\to Sub(J)</annotation></semantics></math> is an equivalence.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>It is a general fact in a regular 2-category that for any eso <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f:X\to Y</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>Sub</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sub</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^*:Sub(Y)\to Sub(X)</annotation></semantics></math> is full (and faithful, which of course is automatic for a functor between posets). For if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>U</mi><mo>≤</mo><msup><mi>f</mi> <mo>*</mo></msup><mi>V</mi></mrow><annotation encoding="application/x-tex">f^*U \le f^*V</annotation></semantics></math>, then we have a square</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><msup><mi>f</mi> <mo>*</mo></msup><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>f</mi> <mo>*</mo></msup><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{f^*U & \to & f^*V & \to & V\\ \downarrow &&&& \downarrow\\ U & & \to & & Y} </annotation></semantics></math></div> <p>in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>U</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">f^*U \to U</annotation></semantics></math> is eso and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">V\to Y</annotation></semantics></math> is ff, hence we have <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\to V</annotation></semantics></math> over <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>.</p> <p>Thus, in our case, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">j^*</annotation></semantics></math> is full and faithful since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> is eso, so it suffices to show that for any ff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">U\to J</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mo>*</mo></msup><msub><mo>∃</mo> <mi>j</mi></msub><mi>U</mi><mo>≤</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">j^* \exists_j U \le U</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sub</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sub(J)</annotation></semantics></math>. But we have a commutative square</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>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mo>∃</mo> <mi>j</mi></msub><mi>U</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>J</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{U & \to & \exists_j U\\ \downarrow && \downarrow\\ J & \to & X} </annotation></semantics></math></div> <p>where the vertical arrows are ff and the bottom arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">J\to X</annotation></semantics></math> is faithful and pseudomonic, from which it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><msub><mo>∃</mo> <mi>j</mi></msub><mi>U</mi></mrow><annotation encoding="application/x-tex">U\to \exists_j U</annotation></semantics></math> is also faithful and pseudomonic. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><msub><mo>∃</mo> <mi>j</mi></msub><mi>U</mi></mrow><annotation encoding="application/x-tex">U\to \exists_j U</annotation></semantics></math> is eso by definition, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is a core of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∃</mo> <mi>j</mi></msub><mi>U</mi></mrow><annotation encoding="application/x-tex">\exists_j U</annotation></semantics></math>, and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mo>*</mo></msup><msub><mo>∃</mo> <mi>j</mi></msub><mi>U</mi></mrow><annotation encoding="application/x-tex">j^*\exists_j U</annotation></semantics></math> is a groupoid mapping to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∃</mo> <mi>j</mi></msub><mi>U</mi></mrow><annotation encoding="application/x-tex">\exists_j U</annotation></semantics></math>, it factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>, as desired.</p> </div> </body></html> </div> <div class="revisedby"> <p> Last revised on February 13, 2009 at 03:36:05. 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