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There are however different formalizations of this idea. Here discussed is the notion, usually going by this term, where an 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> is called <em>compact</em> if mapping out of it <a class="existingWikiWord" href="/nlab/show/preserved+colimit">preserves</a> <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a>.</p> <p>This means that if any other 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> is given as the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of a “suitably increasing” family of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{A_i\}</annotation></semantics></math>, then every morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><mi>A</mi><mo>=</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mi>i</mi></munder></munder><msub><mi>A</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> X \longrightarrow A = \underset{\underset{i}{\longrightarrow}}{\lim} A_i </annotation></semantics></math></div> <p>out of the compact 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> into that colimit factors through one of the <a class="existingWikiWord" href="/nlab/show/coprojections">coprojections</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>i</mi></msub><mo>→</mo><munder><mrow><mi>lim</mi><msub><mi>A</mi> <mi>i</mi></msub></mrow><munder><mo>⟶</mo><mi>i</mi></munder></munder></mrow><annotation encoding="application/x-tex">A_i \to \underset{\underset{i}{\longrightarrow}}{lim A_i}</annotation></semantics></math>.</p> <p>The notion of <em><a class="existingWikiWord" href="/nlab/show/small+object">small object</a></em> is essentially the same, with a bit more flexibility on when the family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{A_i\}</annotation></semantics></math> is taken to be “suitably increasing”. An important application of the above factorization property is accordingly named the <em><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></em>. On the other hand, there is also the notion of <em><a class="existingWikiWord" href="/nlab/show/finite+object">finite object</a></em> (in a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>) which, while closely related, is different. See also <em><a href="#SubtletiesAndDifferentMeanings">Subtleties and different meanings</a></em> below.</p> <h2 id="FinitelyPresentableObject">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> that admits <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a>. Then 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><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X \in C</annotation></semantics></math> is <strong>compact</strong>, or <strong>finitely presented</strong> or <strong>finitely presentable</strong>, or <strong>of finite presentation</strong>, if the <a class="existingWikiWord" href="/nlab/show/corepresentable+functor">corepresentable functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> Hom_C(X,-) \colon C \to Set </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> these <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a>. This means that for every <a class="existingWikiWord" href="/nlab/show/filtered+category">filtered category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> and every functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D \to 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><munder><mi>lim</mi><munder><mo>→</mo><mi>d</mi></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><munder><mi>lim</mi><munder><mo>→</mo><mi>d</mi></munder></munder><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underset{\underset{d}{\to}}{\lim} C(X,F(d)) \xrightarrow{\simeq} C(X, \underset{\underset{d}{\to}}{\lim} F(d)) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p>More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/regular+cardinal">regular cardinal</a>, then an 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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(X,-)</annotation></semantics></math> commutes with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a> is called <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-compact</strong>, or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-presented</strong>, or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-presentable</strong>. An object which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-compact for some regular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> is called a <a class="existingWikiWord" href="/nlab/show/small+object">small object</a>.</p> </div> <h2 id="Properties">Properties</h2> <p> <div class='num_prop' id='SmoothColimitsOfCompactObjectsAreCompact'> <h6>Proposition</h6> <p><strong>(smooth colimits of compact objects are compact)</strong> <br /> A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-small <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-compact objects is again a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-compact object.</p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X : D \to C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-compact objects. Let <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> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/filtered+category">filtered category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A : I \to C</annotation></semantics></math> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-filtered diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mi>d</mi></munder></munder><msub><mi>X</mi> <mi>d</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟶</mo><mi>i</mi></munder></munder><msub><mi>A</mi> <mi>i</mi></msub><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟵</mo><mi>d</mi></munder></munder><mspace width="thickmathspace"></mspace><mi>Hom</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><msub><mi>X</mi> <mi>d</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟶</mo><mi>i</mi></munder></munder><msub><mi>A</mi> <mi>i</mi></msub><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> Hom \Big( \underset{\underset{d}{\longrightarrow}}{\lim} X_d ,\, \underset{\underset{i}{\longrightarrow}}{\lim} A_i \Big) \;\simeq\; \underset{\underset{d}{\longleftarrow}}{\lim} \; Hom \Big( X_d ,\, \underset{\underset{i}{\longrightarrow}}{\lim} A_i \Big) </annotation></semantics></math></div> <p>by <a class="existingWikiWord" href="/nlab/show/hom-functor+preserves+limits">general properties of</a> the <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a>. Now using that every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">X_d</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-compact and <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-filtered this is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟵</mo><mi>d</mi></munder></munder><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟶</mo><mi>i</mi></munder></munder><mspace width="thickmathspace"></mspace><mi>Hom</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>X</mi> <mi>d</mi></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \;\simeq\; \underset{\underset{d}{\longleftarrow}}{\lim} \; \underset{\underset{i}{\longrightarrow}}{\lim} \; Hom\big(X_d, A_i\big) \,. </annotation></semantics></math></div> <p>Since this (co)limit is taken in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-small limit over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/limits+commuting+with+colimits">commutes</a> with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-filtered colimit</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟶</mo><mi>i</mi></munder></munder><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟵</mo><mi>d</mi></munder></munder><mspace width="thinmathspace"></mspace><mi>Hom</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>X</mi> <mi>d</mi></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \;\simeq\; \underset{\underset{i}{\longrightarrow}}{\lim} \; \underset{\underset{d}{\longleftarrow}}{\lim} \, Hom\big(X_d, A_i\big) \,. </annotation></semantics></math></div> <p>Finally we make take the limit again to a colimit in the first argument</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mi>lim</mi><munder><mo>⟶</mo><mi>i</mi></munder></munder><mspace width="thickmathspace"></mspace><mi>Hom</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mi>d</mi></munder></munder><msub><mi>X</mi> <mi>d</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>A</mi> <mi>i</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \cdots \;\simeq\; \underset{\underset{i}{\longrightarrow}}{\lim} \; Hom \big( \underset{\underset{d}{\longrightarrow}}{\lim} X_d ,\, A_i \big) \,, </annotation></semantics></math></div> <p>which yields the claim.</p> </div> </p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>In a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/accessible+category">accessible category</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/sharply+smaller+cardinal">sharply smaller</a> than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>, then every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-compact object is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-filtered colimit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-compact objects.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>See <a href="#AdamekRosicky94">Adamek-Rosicky, Remark 2.15</a>. It is noted there that with the more technical proof of <a href="#MakkaiPare89">Makkai-Pare, Proposition 2.3.11</a> the words “a retract of” can be omitted.</p> </div> <p>If we weaken the hypothesis to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>≤</mo><mi>κ</mi></mrow><annotation encoding="application/x-tex">\lambda\le \kappa</annotation></semantics></math>, then we retain all of the result except for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-filteredness of the colimit.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>In a locally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/locally+presentable+category">presentable category</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>≤</mo><mi>κ</mi></mrow><annotation encoding="application/x-tex">\lambda\le \kappa</annotation></semantics></math>, then every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-compact object is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-small colimit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-compact objects.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>See <a href="#AdamekRosicky94">Adamek-Rosicky, Remark 1.30</a>. It is claimed there that the words “a retract of” can be omitted by reference to an argument in Makkai-Pare, but it seems unclear how this argument is intended to be used. An alternative proof of this improvement is proposed <a href="https://mathoverflow.net/q/325278">at this mathoverflow question</a>.</p> </div> <h2 id="Examples">Examples</h2> <ul> <li> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> an object is compact precisely if it is a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a>. For this to hold constructively, <a class="existingWikiWord" href="/nlab/show/filtered+categories">filtered categories</a> (appearing in the definition of <em><a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></em>) have to be understood as categories admitting cocones of every <em>Bishop-finite</em> diagram. (An object of Set is a <a class="existingWikiWord" href="/nlab/show/finite+set">Kuratowski-finite</a> precisely if it is a <a class="existingWikiWord" href="/nlab/show/finitely+generated+object">finitely generated object</a>, or equivalently if it is <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> when regarded as a <a class="existingWikiWord" href="/nlab/show/discrete+object">discrete</a> topological space.)</p> </li> <li> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/topos">topos</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 compact if:</p> <ul> <li> <p><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 <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos"> sheaf topos</a> on a site whose topology is generated by finite covering families and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a representable sheaf;</p> </li> <li> <p>in particular 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 <a class="existingWikiWord" href="/nlab/show/coherent+topos">coherent topos</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/coherent+object">coherent object</a>.</p> </li> </ul> </li> </ul> <p>However, there exist compact objects which are not coherent, c.f. the <a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a>, D3.3.12.</p> <ul> <li> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> an object is compact precisely if it is <a class="existingWikiWord" href="/nlab/show/finitely+presented+group">finitely presented</a> as a group.</p> </li> <li> <p>More generally, 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 any <a class="existingWikiWord" href="/nlab/show/variety+of+algebras">variety of algebras</a>, then an object is compact precisely if it is <a class="existingWikiWord" href="/nlab/show/finitely+presented+algebra">finitely presented</a> as an algebra. A proof may be found in <a href="#AdamekRosicky94">Adámek-Rosický 94, Corollary 3.13</a>.</p> </li> <li> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = Op(X)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</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>. Then an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">U \in C</annotation></semantics></math> is a compact object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> precisely if it is a <a class="existingWikiWord" href="/nlab/show/compact+space">compact topological space</a>. (It is <em>not</em> true that <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 a compact object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> iff it is a compact topological space; see below.)</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional vector space</a> is compact in <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>, see <a href="finite-dimensional+vector+space#CompactClosure">here</a>.</p> </li> </ul> <h2 id="SubtletiesAndDifferentMeanings">Subtleties and different meanings</h2> <p>One has to be careful about the following variations of the theme of compactness.</p> <p>(Some of these subtleties are resolved by noticing that there is a hierarchy of notions of compact objects that are secretly different but partly go by the same name. Some discussion of this is currently at <em><a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></em>, but more detailed discussion should eventually be somewhere…)</p> <p>In the <a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a>, what Johnstone calls <em>compact objects</em> are those objects such that the <a class="existingWikiWord" href="/nlab/show/top">top</a> element of the <a class="existingWikiWord" href="/nlab/show/poset+of+subobjects">poset of subobjects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Sub</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\operatorname{Sub}(C)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compact+element">compact element</a>; he reserves the term <em>finitely-presented</em> for the notion of compact on this page.</p> <h3 id="CompactnessInAdditiveCategories">Compactness in additive categories</h3> <p>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 an <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a> (often a <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>), an 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">C</annotation></semantics></math> is called <strong>compact</strong> if for every set <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> of objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> such that the coproduct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub><mi>s</mi></mrow><annotation encoding="application/x-tex">\coprod_{s\in S} s</annotation></semantics></math> exists, the canonical map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \coprod_{s\in S} C(x,s)\to C(x,\coprod_{s\in S}s) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoids</a>.</p> <p>Here is an application of this concept to characterize which abelian categories are categories of modules of some ring:</p> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>Let <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> be an abelian category. 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 all <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> and has a compact <a class="existingWikiWord" href="/nlab/show/projective+object"> projective</a> <a class="existingWikiWord" href="/nlab/show/generator">generator</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≃</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">C \simeq R Mod</annotation></semantics></math> for some ring <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>. In fact, in this situation we can take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">R = C(x,x)^{op}</annotation></semantics></math> where <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 any <a class="existingWikiWord" href="/nlab/show/compact+projective+object">compact projective</a> generator. Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≃</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">C \simeq R Mod</annotation></semantics></math>, then <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 all small coproducts and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">x = R</annotation></semantics></math> is a compact projective generator.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This theorem, minus the explicit description 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>, can be found as Exercise F on page 103 of Peter Freyd’s book <a href="http://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf#page=132">Abelian Categories</a>. The first part of this theorem can also be found as Prop. 2.1.7. of Victor Ginzburg’s <a href="http://arxiv.org/PS_cache/math/pdf/0506/0506603v1.pdf#page=4">Lectures on noncommutative geometry</a>. Conversely, it is easy to see that <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 a compact projective generator of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math>.</p> </div> <div class="query"> <p>Zoran: While Ginzburg’s reference is surely a worthy to look at, it would be better not to give false impression that this <a class="existingWikiWord" href="/nlab/show/reconstruction+theorem">reconstruction theorem</a> is due Ginzburg or at all new. It is rather a classical and well know fact probably from early 1960s, essentially small strengthening of a variant of a circle of abelian reconstruction theorems including the <a href="http://myyn.org/m/article/gabriel-popescu-theorem-for-ab5-categories">Gabriel-Popescu theorem</a>(probably our variant could be read off from classical algera book by Faith for example, or Popescu’s book on abelian categories, in any case it is well known in <a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a>). In fact for this fact, if I think better, the reconstruction belongs usually to expositions which treat classical Morita theory for rings.</p> </div> <p>A triangulated category is <strong>compactly generated</strong> if it is generated (see <a class="existingWikiWord" href="/nlab/show/generator">generator</a>) by a <em>set</em> of compact objects.</p> <p>The notion can be modified for categories <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> (compare to the notions of finite and/or rigid objects in various contexts).</p> <p>Compact objects in the derived categories of quasicoherent sheaves over a scheme are called <a class="existingWikiWord" href="/nlab/show/perfect+complexes">perfect complexes</a>. Any <a class="existingWikiWord" href="/nlab/show/compact+object">compact object</a> in the <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> over a perfect ring is finitely generated as a module.</p> <p>In non-additive contexts, the above definition is not right. For instance, with this definition a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> would be compact iff it is <a class="existingWikiWord" href="/nlab/show/connected+space">connected</a>. In general one should expect to instead preserve filtered colimits, as above.</p> <h3 id="CompactObjectsInTop">Compact objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math></h3> <p>Recall the above example of <a class="existingWikiWord" href="/nlab/show/compact+space">compact topological spaces</a>. Notice that the statement which one might expect, that a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</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 <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> if it is a compact object in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, is not quite right in general.</p> <p>A <a class="existingWikiWord" href="/nlab/show/counterexample">counterexample</a> is given for instance in <a href="#Hovey99">Hovey (1999), page 49</a>, which itself was corrected by Don Stanley (see the <a href="http://hopf.math.purdue.edu/Hovey/model-err.pdf">errata</a> of that book). See also the blog discussion <a href="http://golem.ph.utexas.edu/category/2009/05/journal_club_geometric_infinit_3.html#c023790">here</a>.</p> <p>Namely, the two-element set with the <a class="existingWikiWord" href="/nlab/show/indiscrete+topology">indiscrete topology</a> is a compact space <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> for which</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mi>M</mi><mo lspace="verythinmathspace">:</mo><mi>Top</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> Hom(X, -) M\colon Top \rightarrow Top </annotation></semantics></math></div> <p>doesn’t preserve filtered colimits, in fact not even <a class="existingWikiWord" href="/nlab/show/sequential+colimit">colimits of sequences</a> (functors out of the <a class="existingWikiWord" href="/nlab/show/poset">ordered set</a> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>).</p> <p>For example, consider the sequence of spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>=</mo><mo stretchy="false">[</mo><mi>n</mi><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> X_n=[n,\infty) \times \{0,1\} </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/open+sets">open sets</a> are of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo>,</mo><mn>∞</mn><mo stretchy="false">]</mo><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>∪</mo><mo stretchy="false">[</mo><mi>m</mi><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> [n, \infty] \times \{0\} \cup [m,\infty) \times \{1\} </annotation></semantics></math></div> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≥</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \geq n</annotation></semantics></math>), plus the empty set. Define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_n \rightarrow X_{n+1}</annotation></semantics></math> so that it sends a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k, \epsilon)</annotation></semantics></math> to itself if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \gt n</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,\epsilon)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1,\epsilon)</annotation></semantics></math>. This defines a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> F: \mathbb{N} \rightarrow Top </annotation></semantics></math></div> <p>The colimit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">X_\infty</annotation></semantics></math> of this sequence is the two-element set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,1\}</annotation></semantics></math> with the indiscrete topology. However, the identity map on this space does not factor through any of the canonical maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>X</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">X_n \rightarrow X_\infty</annotation></semantics></math>. It follows that the comparison map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>colim</mi><mi>n</mi></munder><mspace width="thickmathspace"></mspace><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>∞</mn></msub><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>∞</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>∞</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underset{n}{colim}\ Hom(X_\infty, X_n) \rightarrow Hom(X_\infty, X_\infty) </annotation></semantics></math></div> <p>is not surjective, and therefore not an isomorphism.</p> <div class="query"> <p><em><a class="existingWikiWord" href="/nlab/show/Todd+Trimble">Todd</a></em> (posted from n-category cafe): I don’t know if the story is any different for <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> compact <em>Hausdorff</em>, but it could be worth considering.</p> </div> <p>But with a bit of care on the assumptions, similar results do hold:</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Y \in Top</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a>, then <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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom(Y,-)</annotation></semantics></math> preserves colimits of functors mapping out of <a class="existingWikiWord" href="/nlab/show/limit+ordinals">limit ordinals</a>, provided that the arrows of the cocone diagram,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>α</mi></msub><mo>→</mo><msub><mi>X</mi> <mi>β</mi></msub><mo>,</mo></mrow><annotation encoding="application/x-tex"> X_\alpha \rightarrow X_\beta, </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/closed+map">closed</a> inclusions of <a class="existingWikiWord" href="/nlab/show/separation+axiom"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>T</mi> <mn>1</mn></msub> </mrow> <annotation encoding="application/x-tex">T_1</annotation> </semantics> </math>-spaces</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">X_\beta</annotation></semantics></math> is the colimit of earlier <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">X_\alpha</annotation></semantics></math> if <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> is itself a limit ordinal.</p> </div> </p> <p>This applies for example to the sequence of inclusions of <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>-skeleta in a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>. By considering the cases <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><msup><mi>S</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">Y=S^k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><msup><mi>S</mi> <mi>k</mi></msup><mo>×</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">Y = S^k \times I</annotation></semantics></math>, this implies that the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\pi_k</annotation></semantics></math> also preserves colimits of such sequences. Applications also occur in the theory of <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying bundles</a>. For example, the total space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>EO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">EO(n)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">O(n)</annotation> </semantics> </math></a>-<a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a> can be constructed as the colimit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>colim</mi><mrow><mi>k</mi><mo>→</mo><mn>∞</mn></mrow></munder><mspace width="thickmathspace"></mspace><msub><mi>V</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{k \to \infty}{colim} \; V_n(\mathbb{R}^{n+k})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/Stiefel+manifolds">Stiefel manifolds</a> of orthonormal <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>-frames in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n+k}</annotation></semantics></math>. To show <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>EO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">EO(n)</annotation></semantics></math> is weakly contractible, it suffices, by this result, to show that for each <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>, the <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_j(V_n(\mathbb{R}^{n+k}))</annotation></semantics></math> vanishes for sufficiently large <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>.</p> <p> <div class='proof'> <h6>Proof</h6> <p>Let <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> denote the colimit of the diagram mapping out of the limit ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>. We must show that every map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon Y \to X</annotation></semantics></math> factors through one of the universal cocone components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>α</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mi>α</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i_\alpha \colon X_\alpha \to X</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo><</mo><mi>κ</mi></mrow><annotation encoding="application/x-tex">\alpha \lt \kappa</annotation></semantics></math>. Suppose not. Starting from any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo><</mo><mi>κ</mi></mrow><annotation encoding="application/x-tex">\alpha \lt \kappa</annotation></semantics></math>, pick a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>∈</mo><mi>f</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_0 \in f(Y)</annotation></semantics></math> not in the image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>α</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>α</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i_\alpha(X_\alpha)</annotation></semantics></math>. There is a minimal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mn>1</mn></msub><mo><</mo><mi>κ</mi></mrow><annotation encoding="application/x-tex">\alpha_1 \lt \kappa</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>∈</mo><msub><mi>i</mi> <mrow><msub><mi>α</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><msub><mi>α</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_0 \in i_{\alpha_1}(X_{\alpha_1})</annotation></semantics></math>. Proceeding inductively, choose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub><mo>∈</mo><mi>f</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_n \in f(Y)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub><mo>∉</mo><msub><mi>i</mi> <mrow><msub><mi>α</mi> <mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><msub><mi>α</mi> <mi>n</mi></msub></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_n \notin i_{\alpha_n}(X_{\alpha_n})</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\alpha_{n+1}</annotation></semantics></math> be the minimal ordinal such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub><mo>∈</mo><msub><mi>i</mi> <mrow><msub><mi>α</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><msub><mi>α</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_n \in i_{\alpha_{n+1}}(X_{\alpha_{n+1}})</annotation></semantics></math>. Then let <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> be the limit of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_n</annotation></semantics></math>, and let <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> be the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_n: n \in \mathbb{N}\}</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊆</mo><msub><mi>i</mi> <mi>β</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>β</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \subseteq i_\beta(X_\beta)</annotation></semantics></math>. Any subset <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is closed, because closure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊆</mo><msub><mi>i</mi> <mi>β</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>β</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K \subseteq i_\beta(X_\beta)</annotation></semantics></math> is equivalent to closure of of each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>∩</mo><msub><mi>i</mi> <mrow><msub><mi>α</mi> <mi>n</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><msub><mi>α</mi> <mi>n</mi></msub></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K \cap i_{\alpha_n}(X_{\alpha_n})</annotation></semantics></math> (since that is how the colimit topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">X_\beta</annotation></semantics></math> is defined), and all these intersections are finite and therefore closed by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math> condition. It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(S)</annotation></semantics></math> is an infinite discrete subspace of <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>, but this contradicts compactness.</p> </div> </p> <p>This proof is essentially that given in <a href="#Hovey99">Hovey (1999), page 50</a>, where this result is needed towards the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> on the way to establishing the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>, see <a href="classical+model+structure+on+topological+spaces#CompactSubsetsAreSmallInCellComplexes">this lemma</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+element">compact element</a> (in a <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+projective+object">compact projective object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+object+in+an+%28%E2%88%9E%2C1%29-category">compact object in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object">small object</a>, <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finitely+generated+object">finitely generated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a>, <a class="existingWikiWord" href="/nlab/show/accessible+category">accessible category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+%28%E2%88%9E%2C1%29-category">compactly generated (∞,1)-category</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/finite+objects">finite objects</a>:</strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/perfect+module">perfect module</a></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/fully+dualizable+object">fully</a>-)<a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/compact+object">compact object</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Compact objects are discussed under the term “finitely presentable” or “finitely-presentable” objects for instance in</p> <ul> <li id="AdamekRosicky94"> <p><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Ad%C3%A1mek">Jiří Adámek</a>, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, <em><a class="existingWikiWord" href="/nlab/show/Locally+Presentable+and+Accessible+Categories">Locally Presentable and Accessible Categories</a></em>, Cambridge University Press in the London Mathematical Society Lecture Note Series, number 189, (1994)</p> </li> <li id="MakkaiPare89"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Makkai">Michael Makkai</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Par%C3%A9">Robert Paré</a>, <em>Accessible categories: The foundations of categorical model theory</em> Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, around Definition 6.3.3 of <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Stone+Spaces">Stone Spaces</a></em>, Definition VI.1.8;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, the <em><a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a></em>, D2.3.1.</p> </li> </ul> <p>For the pages quoted in the context of the discussion of compact objects in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> see</p> <ul> <li id="Hovey99"><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em><a class="existingWikiWord" href="/nlab/show/Model+Categories">Model Categories</a></em>, Mathematical Surveys and Monographs, Volume 63, AMS (1999). <a href="https://bookstore.ams.org/surv-63-s">ISBN:978-0-8218-4361-1</a> <a href="https://doi.org/http://dx.doi.org/10.1090/surv/063">doi:10.1090/surv/063</a> <a href="https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf">pdf</a>, (errata: <a href="https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats-errata.pdf">pdf</a>) <p>.</p> </li> </ul> <p>For the general definition with an eye towards the definition of <a class="existingWikiWord" href="/nlab/show/compact+object+in+an+%28infinity%2C1%29-category">compact object in an (infinity,1)-category</a> see section A.1.1 section 5.3.4 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 29, 2024 at 09:23:54. 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