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href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#StabilityUnderOperations'>Stability under various operations</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <strong>accessible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category</strong> is the generalization of the notion of <a class="existingWikiWord" href="/nlab/show/accessible+category">accessible category</a> from <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> theory.</p> <p>It is a means to handle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories that are not <a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a> in terms of small data.</p> <p>An <em>accessible</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category is one which may be <a class="existingWikiWord" href="/nlab/show/large+category">large</a>, but can entirely be <em>accessed</em> as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category of “conglomerates of objects” in a small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category – precisely: that it is a category 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>-small <a class="existingWikiWord" href="/nlab/show/ind-object">ind-object</a>s in some small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>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>-accessible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category which in addition has all <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimits">(∞,1)-colimits</a> is called a <em><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally ∞-presentable</a></em> or a <em><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/compactly+generated+%28%E2%88%9E%2C1%29-category">compactly generated (∞,1)-category</a></em>.</p> <h2 id="definition">Definition</h2> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/regular+cardinal">regular cardinal</a>.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is <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>-accessible</strong> if it satisfies the following equivalent conditions:</p> <ol> <li> <p>There is a <a class="existingWikiWord" href="/nlab/show/small+%28%E2%88%9E%2C1%29-category">small (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^0</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>≃</mo><msub><mi>Ind</mi> <mi>κ</mi></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C} \simeq Ind_\kappa(C^0) </annotation></semantics></math></div> <p>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">\mathcal{C}</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+ind-objects">(∞,1)-category of ind-objects</a>, relative <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>, in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^0</annotation></semantics></math>.</p> </li> <li> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></p> <ol> <li> <p>is <a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p>has all <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></p> </li> <li> <p>the full <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-category">sub-(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mi>κ</mi></msup><mo>↪</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}^\kappa \hookrightarrow \mathcal{C}</annotation></semantics></math> 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>-<a class="existingWikiWord" href="/nlab/show/compact+objects">compact objects</a> is an <a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small (∞,1)-category</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mi>κ</mi></msup><mo>↪</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}^\kappa \hookrightarrow \mathcal{C}</annotation></semantics></math> generates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> under <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+%28%E2%88%9E%2C1%29-colimits">filtered (∞,1)-colimits</a>.</p> </li> </ol> </li> <li> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></p> <ol> <li> <p>is <a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p>has all <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></p> </li> <li> <p>there is <em>some</em> <a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-category">sub-(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>′</mo><mo>↪</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}' \hookrightarrow \mathcal{C}</annotation></semantics></math> 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>-<a class="existingWikiWord" href="/nlab/show/compact+objects">compact objects</a> which generates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> under <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+%28%E2%88%9E%2C1%29-colimits">filtered (∞,1)-colimits</a>.</p> </li> </ol> </li> </ol> <p>The notion of accessibility is mostly interesting for <em>large</em> (∞,1)-categories. For</p> <ul> <li>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">\mathcal{C}</annotation></semantics></math> is small, then there exists 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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</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>-accessible if and only 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">\mathcal{C}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete (∞,1)-category</a>.</li> </ul> <p>Generally, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is called an <strong>accessible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category</strong> if it 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>-accessible for some regular cardinal <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>.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>These conditions are indeed equivalent.</p> </div> <p>For the first few this is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 5.4.2.2</a>. The last one is in <a class="existingWikiWord" href="/nlab/show/HTT">HTT, section 5.4.3</a>.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>An <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> between accessible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories that preserves <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 colimits is called an <strong><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-functor">accessible (∞,1)-functor</a></strong> .</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>AccCat</mi><mo>⊂</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">(\infty,1)AccCat \subset (\infty,1)Cat</annotation></semantics></math> for the 2-<a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-category">sub-(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a> on</p> <ul> <li> <p>those objects that are accessible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories;</p> </li> <li> <p>those morphisms for which there is 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> such that the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> 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>-continuous and preserves <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/compact+object">compact object</a>s.</p> </li> </ul> </div> <p>So morphisms are the <a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-functor">accessible (∞,1)-functor</a>s that also preserves <a class="existingWikiWord" href="/nlab/show/compact+object">compact object</a>s. (?)</p> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, def. 5.4.2.16</a>.</p> <h2 id="properties">Properties</h2> <h3 id="StabilityUnderOperations">Stability under various operations</h3> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is an accessible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category then so are</p> <ul> <li> <p>for <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> a small simplicial set the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(K,C)</annotation></semantics></math>;</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">p : K \to C</annotation></semantics></math> a small <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>, the <a class="existingWikiWord" href="/nlab/show/over+quasi-category">over quasi-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>p</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{/p}</annotation></semantics></math> and under-quasi-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mi>p</mi><mo stretchy="false">/</mo></mrow></msub></mrow><annotation encoding="application/x-tex">C_{p/}</annotation></semantics></math>.</p> </li> </ul> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT</a> section 5.4.4, 5.4.5 and 5.4.6.</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>The <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> of accessible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a> is again accessible.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, section 5.4.6</a>.</p> <p>Generally:</p> <div class="num_theorem"> <h6 id="theorem_3">Theorem</h6> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>AccCat</mi></mrow><annotation encoding="application/x-tex">(\infty,1)AccCat</annotation></semantics></math> has all small <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a>s and the inclusion</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><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>AccCAT</mi><mo>↪</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>CAT</mi></mrow><annotation encoding="application/x-tex"> (\infty,1)AccCAT \hookrightarrow (\infty,1)CAT </annotation></semantics></math></div> <p>preserves these.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, proposition 5.4.7.3</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/compactly+generated+%28%E2%88%9E%2C1%29-category">compactly generated (∞,1)-category</a></li> </ul> <div> <p><strong>Locally presentable categories:</strong> <a class="existingWikiWord" href="/nlab/show/cocomplete+category">Cocomplete</a> possibly-<a class="existingWikiWord" href="/nlab/show/large+categories">large categories</a> generated under <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a> by <a class="existingWikiWord" href="/nlab/show/small+object">small</a> <a class="existingWikiWord" href="/nlab/show/generators">generators</a> under <a class="existingWikiWord" href="/nlab/show/small+colimit">small</a> <a class="existingWikiWord" href="/nlab/show/relations">relations</a>. Equivalently, <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible</a> <a class="existingWikiWord" href="/nlab/show/reflective+localizations">reflective localizations</a> of <a class="existingWikiWord" href="/nlab/show/free+cocompletions">free cocompletions</a>. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a> localization.</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/toposes">toposes</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th>locally presentable</th><th>loc finitely pres</th><th>localization theorem</th><th><a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a></th><th>accessible</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locales">locales</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/suplattice">suplattice</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+lattices">algebraic lattices</a></td><td style="text-align: left;"><a href="algebraic+lattice#RelationToLocallyFinitelyPresentableCategories">Porst’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/powerset">powerset</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/poset">poset</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+finitely+presentable+categories">locally finitely presentable categories</a></td><td style="text-align: left;"><a href="locally+presentable+category#AsLocalizationsOfPresheafCategories">Gabriel–Ulmer’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+toposes">model toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dugger%27s+theorem">Dugger's theorem</a></td><td style="text-align: left;">global <a class="existingWikiWord" href="/nlab/show/model+structures+on+simplicial+presheaves">model structures on simplicial presheaves</a></td><td style="text-align: left;">n/a</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-toposes">(∞,1)-toposes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-categories">locally presentable (∞,1)-categories</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a href="locally+presentable+infinity-category#Definition">Simpson’s theorem</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf+%28%E2%88%9E%2C1%29-categories">(∞,1)-presheaf (∞,1)-categories</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-categories">accessible (∞,1)-categories</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Theory of <a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible 1-categories</a>:</p> <ul> <li id="AdamekRosicky"><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, (1994)</li> </ul> <p>Theory of accessible <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, Section 5.4 of <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> <p>See also:</p> <ul> <li id="Rezk2021"><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Generalizing accessible ∞-categories</em>, 2021 (<a href="https://faculty.math.illinois.edu/~rezk/accessible-cat-thoughts.pdf">pdf</a>).</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 1, 2021 at 04:46:44. 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