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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="compact_objects">Compact objects</h4> <div class="hide"><div> <p><strong>objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">d \in C</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>d</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(d,-)</annotation></semantics></math> commutes with certain <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coproduct-preserving+representable">coproduct-preserving representable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+object">connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+object">compact object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+element">compact object in a (0,1)-category</a>, <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/finite+object">finite object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object">small object</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tiny+object">tiny object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+object">atomic object</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+category">accessible category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/presentable+category">presentable category</a></li> </ul> </li> </ul> <h2 id="models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a></p> </li> </ul> <h2 id="relative_version">Relative version</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/compact+object+-+contents">Edit this sidebar</a> </p> </div></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='#raising_the_index_of_accessibility'>Raising the index of accessibility</a></li> <li><a href='#stability_under_various_constructions'>Stability under various constructions</a></li> <li><a href='#adjoint_functor_theorem'>Adjoint functor theorem</a></li> <li><a href='#idempotence_completeness'>Idempotence completeness</a></li> <li><a href='#categories_of_models_over_a_theory'>Categories of models over a theory</a></li> <li><a href='#wellpoweredness_and_wellcopoweredness'>Well-poweredness and well-copoweredness</a></li> <li><a href='#TheTwoCategoryOfAccessibleCategories'>The 2-category of accessible categories</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#functor_categories'>Functor categories</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesInEnrichedCategoryTheory'>In enriched category theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>An accessible <a class="existingWikiWord" href="/nlab/show/category">category</a> is a possibly <a class="existingWikiWord" href="/nlab/show/large+category">large category</a> which is however essentially determined by a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>, in a certain way.</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>. Recall an object <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: \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>-compact iff <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">\mathcal{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>-filtered colimits.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small 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:</p> <ol> <li> <p>the category has <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/directed+colimits">directed colimits</a> (or, equivalently, <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), and</p> </li> <li> <p>there is a <a class="existingWikiWord" href="/nlab/show/set">set</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>-<a class="existingWikiWord" href="/nlab/show/compact+objects">compact objects</a> that generate the category 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>-directed colimits.</p> </li> </ol> <p>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">\mathcal{C}</annotation></semantics></math> is an <strong>accessible category</strong> if 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 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.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Unlike for <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a>, it does not follow that 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 <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 and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo><</mo><mi>λ</mi></mrow><annotation encoding="application/x-tex">\kappa\lt \lambda</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">\mathcal{C}</annotation></semantics></math> is also <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>-accessible. It is true, however, that for any accessible category, there are arbitrarily large cardinals <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>-accessible.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Equivalent characterizations include that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is accessible iff:</p> <ul> <li> <p>it is the category of <a class="existingWikiWord" href="/nlab/show/models">models</a> (in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>) of some small <a class="existingWikiWord" href="/nlab/show/sketch">sketch</a>.</p> </li> <li> <p>it is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">Ind</mi> <mi>κ</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Ind}_\kappa(S)</annotation></semantics></math> for <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> small, i.e. 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>-<a class="existingWikiWord" href="/nlab/show/ind-object">ind-completion</a> of a small category, for some <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> </li> <li> <p>it is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mspace width="thinmathspace"></mspace><mi mathvariant="normal">Flat</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\kappa\,\mathrm{Flat}(S)</annotation></semantics></math> for <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> small and some <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>, i.e. the 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>-<a class="existingWikiWord" href="/nlab/show/flat+functor">flat</a> functors from some small category to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>.</p> </li> <li> <p>it is the category of models (in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Set</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Set}</annotation></semantics></math>) of a suitable type of logical theory.</p> </li> </ul> </div> <p>The relevant notion of <a class="existingWikiWord" href="/nlab/show/functor">functor</a> between accessible categories is</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F\colon C\to D</annotation></semantics></math> between accessible categories is an <strong><a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible functor</a></strong> if 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">C</annotation></semantics></math> and <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> are both <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 and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> preserves <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> </div> <h2 id="Properties">Properties</h2> <h3 id="raising_the_index_of_accessibility">Raising the index of accessibility</h3> <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">\mathcal{C}</annotation></semantics></math> is <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>-accessible and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>⊴</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\lambda\unlhd\mu</annotation></semantics></math> (see <a class="existingWikiWord" href="/nlab/show/sharply+smaller+cardinal">sharply smaller cardinal</a>), 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">\mathcal{C}</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>-accessible. Thus, any accessible category is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>-accessible for arbitrarily large cardinals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>.</p> <h3 id="stability_under_various_constructions">Stability under various constructions</h3> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is an accessible category and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>, then the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>K</mi> <mi>op</mi></msup><mo>,</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(K^{op}, \mathcal{C})</annotation></semantics></math> is again accessible.</p> </div> <p>(<a href="#Lurie">Lurie, prop. 5.4.4.3</a>)</p> <div class="num_prop" id="StabilityUnderInverseImage"> <h6 id="proposition_3">Proposition</h6> <p><strong>(preservation of accessibility under inverse images)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F : \mathcal{C} \to \mathcal{D}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> between <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a> which 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/filtered+category">filtered</a> <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒟</mi> <mn>0</mn></msub><mo>⊂</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}_0 \subset \mathcal{D}</annotation></semantics></math> be an accessible subcategory. Then the inverse image <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><msub><mi>𝒟</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">f^{-1}(\mathcal{D}_0) \subset C</annotation></semantics></math> 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>-accessible subcategory.</p> </div> <p>This appears as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, corollary A.2.6.5</a>.</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p><strong>(accessibility of fibrations and weak equivalences in a combinatorial model category)</strong></p> <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">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Arr</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr(\mathcal{C})</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>⊂</mo><mi>Arr</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \subset Arr(\mathcal{C})</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> on the weak equivalences and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⊂</mo><mi>Arr</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \subset Arr(\mathcal{C})</annotation></semantics></math> the full subcategory on the fibrations. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∩</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">F \cap W</annotation></semantics></math> are accessible subcategories of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Arr</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr(\mathcal{C})</annotation></semantics></math>.</p> </div> <p>This appears as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, corollary A.2.6.6</a>.</p> <h3 id="adjoint_functor_theorem">Adjoint functor theorem</h3> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p><strong>(adjoint functors)</strong></p> <p>Every accessible functor satisfies the <a class="existingWikiWord" href="/nlab/show/solution+set+condition">solution set condition</a>, and every left or right adjoint between accessible categories is accessible. Therefore, the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a> takes an especially pleasing form for accessible categories that are complete and cocomplete (i.e. are <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable</a>): a functor between such categories is a left (resp. right) adjoint iff it is accessible and preserves all small colimits (resp. limits).</p> </div> <h3 id="idempotence_completeness">Idempotence completeness</h3> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> is accessible precisely when it is <a class="existingWikiWord" href="/nlab/show/idempotent+complete+category">idempotent complete</a>.</p> </div> <p><a href="#MakkaiParé1989">Makkai & Paré (1989)</a> say that this means accessibility is an “almost pure smallness condition.”</p> <h3 id="categories_of_models_over_a_theory">Categories of models over a theory</h3> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p>For a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}</annotation></semantics></math> the following are equivalent:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}</annotation></semantics></math> is finitely <a class="existingWikiWord" href="/nlab/show/accessible+category">accessible</a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo>≃</mo><mi>𝕋</mi><mtext>-</mtext><mi>Mod</mi><mo stretchy="false">(</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}\simeq \mathbb{T}\text{-}Mod(Set)</annotation></semantics></math> for some <a class="existingWikiWord" href="/nlab/show/theory+of+presheaf+type">theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>𝕋</mi> </mrow> <annotation encoding="application/x-tex">\mathbb{T}</annotation> </semantics> </math> of presheaf type</a>.</p> </li> </ul> </div> <p>Moreover, one has the following result due to <a class="existingWikiWord" href="/nlab/show/Christian+Lair">Christian Lair</a>:</p> <div class="num_prop"> <h6 id="proposition_8">Proposition</h6> <p>For a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}</annotation></semantics></math> the following are equivalent:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/accessible+category">accessible</a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/sketch">sketchable</a>.</p> </li> </ul> </div> <p>See also at <em><a href="model+theory#CategoricalModelTheory">categorical model theory</a></em>.</p> <h3 id="wellpoweredness_and_wellcopoweredness">Well-poweredness and well-copoweredness</h3> <ul> <li> <p>Every accessible 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> is <a class="existingWikiWord" href="/nlab/show/well-powered+category">well-powered</a>, since it has a small <a class="existingWikiWord" href="/nlab/show/dense+subcategory">dense subcategory</a> <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">\mathcal{A}</annotation></semantics></math>, for which the <a class="existingWikiWord" href="/nlab/show/restricted+Yoneda+embedding">restricted Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>𝒜</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} \to [\mathcal{A}^{op},Set]</annotation></semantics></math> is fully faithful and preserves monomorphisms, hence embeds the subobject posets 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> as sub-posets of those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>𝒜</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{A}^{op},Set]</annotation></semantics></math>.</p> </li> <li> <p>Every accessible category with <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a> is well-<em>copowered</em>. This is shown in <a href="#AdámekRosický1994">Adamek-Rosicky, Proposition 1.57 and Theorem 2.49</a>. Whether this is true for all accessible categories depends on what <a class="existingWikiWord" href="/nlab/show/large+cardinal">large cardinal</a> properties hold: by Corollary 6.8 of Adamek-Rosicky, if <a class="existingWikiWord" href="/nlab/show/Vopenka%27s+principle">Vopenka's principle</a> holds then all accessible categories are well-copowered, while by Example A.19 of Adamek-Rosicky, if all accessible categories are well-copowered then there exist arbitrarily large <a class="existingWikiWord" href="/nlab/show/measurable+cardinals">measurable cardinals</a>.</p> </li> </ul> <h3 id="TheTwoCategoryOfAccessibleCategories">The 2-category of accessible categories</h3> <p>Write <a class="existingWikiWord" href="/nlab/show/AccCat">AccCat</a> for the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/accessible+functors">accessible functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a>.</p> </li> </ul> <p> <div class='num_prop' id='AccCatHasAllPIELimits'> <h6>Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/AccCat">AccCat</a> has all (lax) <a class="existingWikiWord" href="/nlab/show/2-limits">2-limits</a> and these are <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved</a> by the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AccCat</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">AccCat \to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>.</p> </div> This appears as <a href="#MakkaiParé1989">Makkai & Paré (1989), Thm. 5.1.6, Cor. 5.1.8</a>, <a href="#AdámekRosický1994">Adámek & Rosický (1994) around Thm. 2.77</a>.</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/cosmos+for+enrichment">cosmos for enrichment</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math> which is (<a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> and) <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable</a>, then the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/AccCat">AccCat</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> <a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a> has all <a class="existingWikiWord" href="/nlab/show/PIE+2-limits">PIE 2-limits</a> and <a class="existingWikiWord" href="/nlab/show/split+idempotent">splittings</a> of <a class="existingWikiWord" href="/nlab/show/idempotent">idempotent</a> <a class="existingWikiWord" href="/nlab/show/equivalence+in+a+2-category">equivalences</a> (equivalently it has all <a class="existingWikiWord" href="/nlab/show/flexible+2-limits">flexible 2-limits</a>), as well as <a class="existingWikiWord" href="/nlab/show/2-pullbacks">2-pullbacks</a> along <a class="existingWikiWord" href="/nlab/show/isofibrations">isofibrations</a>.</p> <p>The analogous statements holds for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math>-enriched and <em><a class="existingWikiWord" href="/nlab/show/conical+limit">conically</a></em> accessible categories, in which case the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mtext>-</mtext><mi>ConAccCat</mi><mo>→</mo><mi>𝒱</mi><mtext>-</mtext><mi>Cat</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}\text{-}ConAccCat \to \mathcal{V}\text{-}Cat</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> these <a class="existingWikiWord" href="/nlab/show/2-limits">2-limits</a>.</p> </div> This is <a href="#LackTendas23">Lack & Tendas (2023), Thm. 5.5, Thm. 5.9</a>.</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p><strong>(directed unions)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/AccCat">AccCat</a> has <a class="existingWikiWord" href="/nlab/show/directed+colimits">directed</a> <a class="existingWikiWord" href="/nlab/show/2-colimits">2-colimits</a> of systems of <a class="existingWikiWord" href="/nlab/show/fully+faithful+functors">fully faithful functors</a>. If there is a proper class of <a class="existingWikiWord" href="/nlab/show/strongly+compact+cardinals">strongly compact cardinals</a>, then it has directed colimits of systems of <a class="existingWikiWord" href="/nlab/show/faithful+functors">faithful functors</a>.</p> </div> </p> <p><a href="#ParéRosický13">Paré & Rosický (2013)</a></p> <h2 id="examples">Examples</h2> <ul> <li>Every small <a class="existingWikiWord" href="/nlab/show/discrete+category">discrete category</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>-accessible for every <a class="existingWikiWord" href="/nlab/show/regular+cardinal">regular cardinal</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>, since every discrete filtered diagram is trivial.</li> </ul> <h3 id="functor_categories">Functor categories</h3> <p>See at <em><a href="functor+category#LocalPresentability">Functor category – Accessibility</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/class-accessible+category">class-accessible 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> <h3 id="general">General</h3> <p>The term <em>accessible category</em> is due to</p> <ul> <li id="MakkaiParé1989"><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 <strong>104</strong>, American Mathematical Society, (1989) [<a href="https://bookstore.ams.org/conm-104">ISBN:978-0-8218-7692-3</a>]</li> </ul> <p>Further monographs (with focus on <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a>):</p> <ul> <li id="AdámekRosický1994"><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>, London Mathematical Society Lecture Note Series <strong>189</strong>, Cambridge University Press (1994) [<a href="https://doi.org/10.1017/CBO9780511600579">doi:10.1017/CBO9780511600579</a>]</li> </ul> <p>See also</p> <ul> <li id="ParéRosický13"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Par%C3%A9">Robert Paré</a>, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, <em>Colimits of accessible categories</em>, Math. Proc. Cambr. Phil. Soc. <strong>155</strong> (2013) 47-50 [<a href="https://doi.org/10.1017/S0305004113000030">doi:10.1017/S0305004113000030</a>, <a href="http://arxiv.org/abs/1110.0767">arXiv:1110.0767</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Ad%C3%A1mek">Jiří Adámek</a>, <a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Lack">Stephen Lack</a>, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, <em>A classification of accessible categories,</em> Journal of Pure and Applied Algebra 175:7-30, 2002. <a href="http://maths.mq.edu.au/~slack/papers/acc.html">abstract</a></p> </li> </ul> <p>which further stratifies the accessible categories in terms of <a class="existingWikiWord" href="/nlab/show/sound+doctrines">sound doctrines</a>.</p> <p>The concept is studied in a 2-categorical setting in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Bourke">John Bourke</a>, <em>Accessible aspects of 2-category theory</em> , arXiv:2003.06375 (2020). (<a href="https://arxiv.org/abs/2003.06375">abstract</a>)</li> </ul> <p>A discussion of <a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-categories">accessible (∞,1)-categories</a> is in <a href="http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=341">section 5.4, p. 341</a> of</p> <ul> <li id="Lurie"><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> <p>Some recent developments in the theory of accessible categories can be found in a series of papers on <a href="model+theory#CategoricalModelTheory">categorical model theory</a> and <a class="existingWikiWord" href="/nlab/show/abstract+elementary+classes">abstract elementary classes</a> (many of which also contain some results about arbitrary accessible categories), such as:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tibor+Beke">Tibor Beke</a>, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, <em>Abstract elementary classes and accessible categories</em>, 2011, <a href="https://arxiv.org/abs/1005.2910">arxiv</a></p> </li> <li> <p>Michael Lieberman, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, Sebastien Vasey, <em>Internal sizes in μ-abstract elementary classes</em>, <a href="https://arxiv.org/abs/1708.06782">arxiv</a></p> </li> <li> <p>Michael Lieberman, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, Sebastien Vasey , <em>Set-theoretic aspects of accessible categories</em>, <a href="https://arxiv.org/abs/1902.06777">arxiv</a></p> </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> <h3 id="ReferencesInEnrichedCategoryTheory">In enriched category theory</h3> <p>Discussion of accessible <a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> (see also <a href="locally+presentable+category#ReferencesInEnrichedCategoryTheory">references on enriched locally presentable categories</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <a class="existingWikiWord" href="/nlab/show/Carmen+Quinteriro">Carmen Quinteriro</a>, <em>Enriched accessible categories</em>, Bull. Austral. Math. Soc. <strong>54</strong> (1996) 489-501 [<a href="https://doi.org/10.1017/S0004972700021900">doi:10.1017/S0004972700021900</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <a class="existingWikiWord" href="/nlab/show/Carmen+Quinteriro">Carmen Quinteriro</a>, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, <em>A theory of enriched sketches</em>, Theory and Applications of Categories, <strong>4</strong> 3 (1998) 47-72 [<a href="http://www.tac.mta.ca/tac/volumes/1998/n3/4-03abs.html">tac:4-03</a>, <a href="http://www.tac.mta.ca/tac/volumes/1998/n3/n3.pdf">pdf</a>]</p> </li> <li id="LackTendas23"> <p><a class="existingWikiWord" href="/nlab/show/Stephen+Lack">Stephen Lack</a>, <a class="existingWikiWord" href="/nlab/show/Giacomo+Tendas">Giacomo Tendas</a>, <em>Virtual concepts in the theory of accessible categories</em>, Journal of Pure and Applied Algebra <strong>227</strong> 2 (2023) 107196 [<a href="https://doi.org/10.1016/j.jpaa.2022.107196">arXiv:10.1016/j.jpaa.2022.107196</a>, <a href="https://arxiv.org/abs/2205.11056">arXiv:2205.11056</a>]</p> <blockquote> <p>New characterizations of (enriched) accessible categories and introducing the notions of virtual orthogonality and virtual reflectivity.</p> </blockquote> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 3, 2023 at 13:31:04. See the <a href="/nlab/history/accessible+category" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/accessible+category" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/620/#Item_37">Discuss</a><span class="backintime"><a href="/nlab/revision/accessible+category/68" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/accessible+category" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/accessible+category" accesskey="S" class="navlink" id="history" rel="nofollow">History (68 revisions)</a> <a href="/nlab/show/accessible+category/cite" style="color: black">Cite</a> <a href="/nlab/print/accessible+category" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/accessible+category" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>