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</li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutativity+of+limits+and+colimits'>commutativity of limits and colimits</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+limit'>small limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/filtered+colimit'>filtered colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/directed+colimit'>directed colimit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/sequential+limit'>sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sifted+colimit'>sifted colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+limit'>connected limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/wide+pullback'>wide pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/preserved+limit'>preserved limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/reflected+limit'>reflected limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/created+limit'>created limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a>, <a class='existingWikiWord' href='/nlab/show/diff/pullback'>fiber product</a>, <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a>, <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a>, <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a>, <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a>, <a class='existingWikiWord' href='/nlab/show/diff/cobase+change'>cobase change</a>, <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a>, <a class='existingWikiWord' href='/nlab/show/diff/coequalizer'>coequalizer</a>, <a class='existingWikiWord' href='/nlab/show/diff/join'>join</a>, <a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a>, <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>, <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a>, <a class='existingWikiWord' href='/nlab/show/diff/direct+product'>direct product</a>, <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+limit'>finite limit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+extension'>Yoneda extension</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end and coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fibered+limit'>fibered limit</a></p> </li> </ul> <h2 id='2categorical'>2-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-limit'>2-limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inserter'>inserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/isoinserter'>isoinserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equifier'>equifier</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inverter'>inverter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/PIE-limit'>PIE-limit</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-pullback'>2-pullback</a>, <a class='existingWikiWord' href='/nlab/show/diff/comma+object'>comma object</a></p> </li> </ul> <h2 id='1categorical_2'>(∞,1)-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-pullback'>(∞,1)-pullback</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id='modelcategorical'>Model-categorical</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+Kan+extension'>homotopy Kan extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+product'>homotopy product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equalizer'>homotopy equalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>homotopy fiber</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+totalization'>homotopy totalization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy end</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coproduct'>homotopy coproduct</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coequalizer'>homotopy coequalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>homotopy cofiber</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+pushout'>homotopy pushout</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+realization'>homotopy realization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href='/nlab/edit/infinity-limits+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='creation_of_limits'>Creation of limits</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#examples'>Examples</a></li><li><a href='#terminological_remarks'>Terminological remarks</a><ul><li><a href='#creation_of_nonexisting_limits'>Creation of non-existing limits</a></li><li><a href='#strictness'>Strictness</a></li></ul></li><li><a href='#remarks'>Remarks</a></li><li><a href='#related_pages'>Related pages</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>Let <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_1' xmlns='http://www.w3.org/1998/Math/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> be a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> and <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mo lspace='verythinmathspace'>:</mo><mi>I</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>J\colon I\to C</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a>, and suppose that the composite <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>∘</mo><mi>J</mi></mrow><annotation encoding='application/x-tex'>F \circ J</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>. We say that <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> <strong>creates</strong> this limit if <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> has a limit, and <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> both <a class='existingWikiWord' href='/nlab/show/diff/preserved+limit'>preserves</a> and <a class='existingWikiWord' href='/nlab/show/diff/reflected+limit'>reflects</a> limits of <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math>. The latter two conditions together mean that a <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a> over <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a limiting cone if and only if its image in <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is a limiting cone over <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>∘</mo><mi>J</mi></mrow><annotation encoding='application/x-tex'>F\circ J</annotation></semantics></math>.</p> <p>Of course, a functor <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> creates a <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> if <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>F</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>F^{op}</annotation></semantics></math> creates the corresponding limit.</p> <p>If <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> creates all limits or colimits of a given type (i.e. over a given category <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>), we simply say that <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> creates that sort of limit (e.g. <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> creates products, <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> creates equalizers, etc.).</p> <h2 id='examples'>Examples</h2> <p>\begin{proposition}\label{MonadicFunctorsCreateLimits} <strong>(<a class='existingWikiWord' href='/nlab/show/diff/monadic+functor'>monadic functors</a> <a class='existingWikiWord' href='/nlab/show/diff/created+limit'>create limits</a>)</strong> A <a class='existingWikiWord' href='/nlab/show/diff/monadic+functor'>monadic functor</a></p> <ol> <li> <p>creates all limits that exist in its <a class='existingWikiWord' href='/nlab/show/diff/target'>codomain</a>;</p> </li> <li> <p>creates all colimits that exist in its codomain and are preserved by the corresponding monad (or, equivalently, by the monadic functor itself).</p> </li> </ol> <p>\end{proposition} (e.g. <a href='#MacLane71'>MacLane 71, Exercise IV.2.2 (p. 138)</a>)</p> <p>Creation of a particular sort of <a class='existingWikiWord' href='/nlab/show/diff/split+coequalizer'>split coequalizer</a> figures prominently in Beck’s <a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a>.</p> <h2 id='terminological_remarks'>Terminological remarks</h2> <h3 id='creation_of_nonexisting_limits'>Creation of non-existing limits</h3> <p>It seems that the notion of “creating a limit” is used most frequently when the limits exist in the codomain. One may want to extend the terminology to cases when such limits don’t exist, which would require making a choice about whether a non-existing limit should be regarded as “created”.</p> <p>In <a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories Work</a> the convention is that a functor creates <em>all</em> limits that do not exist in its codomain. In this case, the more generally applicable definition could be stated as “<math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> creates limits for <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> if <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> has a limit whenever <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>∘</mo><mi>J</mi></mrow><annotation encoding='application/x-tex'>F\circ J</annotation></semantics></math> has a limit, and <em>in that case</em> limits of <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> are preserved and reflected by <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>.” (But see below for an additional difference with <a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories Work</a>.)</p> <p>On the other hand, one might argue that it doesn’t make sense to regard a limit that exists in the domain as being “created by the functor” if the limit in the codomain doesn’t even exist. In this case the more generally applicable definition could be stated as “<math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> creates limits for <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> if <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> has a limit whenever <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>∘</mo><mi>J</mi></mrow><annotation encoding='application/x-tex'>F\circ J</annotation></semantics></math> has a limit, and <em>furthermore in all cases</em> limits of <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> are preserved and reflected by <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>.”</p> <p>Finally, one might even argue that based on the meaning of the English word “created”, only something that exists can be created at all. In this case the more generally applicable definition could be stated as “<math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> creates limits for <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> if <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>∘</mo><mi>J</mi></mrow><annotation encoding='application/x-tex'>F\circ J</annotation></semantics></math> both have limits, and furthermore limits of <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> are preserved and reflected by <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>.”</p> <h3 id='strictness'>Strictness</h3> <p>The definitions given above are all “up to isomorphism”, i.e. they satisfy the <a class='existingWikiWord' href='/nlab/show/diff/principle+of+equivalence'>principle of equivalence</a>. The definition in <a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories Work</a> is additionally <em>strict</em>: it requires that for every limiting cone <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> over <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mi>J</mi></mrow><annotation encoding='application/x-tex'>F J</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> there exists a <em>unique</em> cone <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>L'</annotation></semantics></math> over <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> which is mapped <em>exactly</em> to <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>, and this <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>L'</annotation></semantics></math> is a limit of <math class='maruku-mathml' display='inline' id='mathml_771e1560e898c664f0feff6d4864e0145b0d7d3e_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math>. This is used in stating the version of the <a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a> that characterizes the <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Moore+category'>category of algebras for a monad</a> up to <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> rather than <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalence of categories</a>. For <a class='existingWikiWord' href='/nlab/show/diff/amnestic+functor'>amnestic</a> <a class='existingWikiWord' href='/nlab/show/diff/isofibration'>isofibrations</a> the strict and the non-strict notion are equivalent.</p> <h2 id='remarks'>Remarks</h2> <p><a href='http://permalink.gmane.org/gmane.science.mathematics.categories/6644'>Kissinger</a> suggested a concise way to state creation/preservation/etc. of limits. However, there is <a href='https://nforum.ncatlab.org/discussion/7024/lifted-limit/?Focus=56765#Comment_56765'>some dispute</a> about its correctness.</p> <h2 id='related_pages'>Related pages</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/preserved+limit'>preserved limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflected+limit'>reflected limit</a></p> </li> <li> <p><strong>created limit</strong></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lifted+limit'>lifted limit</a></p> </li> </ul> <h2 id='references'>References</h2> <ul> <li id='MacLane71'> <p><a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders Mac Lane</a>, Definition V.1 in: <em><a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories for the Working Mathematician</a></em> (1971)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Ji%C5%99%C3%AD+Ad%C3%A1mek'>Jiří Adámek</a>, <a class='existingWikiWord' href='/nlab/show/diff/Horst+Herrlich'>Horst Herrlich</a>, <a class='existingWikiWord' href='/nlab/show/diff/George+Strecker'>George E. Strecker</a>, Definition 13.17(2) in: <em><a class='existingWikiWord' href='/nlab/show/diff/Abstract+and+Concrete+Categories'>Abstract and Concrete Categories</a></em>.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, §3.3 in: <em><a class='existingWikiWord' href='/nlab/show/diff/Category+Theory+in+Context'>Category Theory in Context</a></em>, Dover Publications (2017) [[pdf](https://math.jhu.edu/~eriehl/context.pdf), <a href='https://math.jhu.edu/~eriehl/context/'>book website</a>]</p> </li> </ul> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on November 7, 2023 at 10:11:10. See the <a href="/nlab/history/created+limit" 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/created+limit" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/12682/#Item_10">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/created+limit/20" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/created+limit" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/created+limit" accesskey="S" class="navlink" id="history" rel="nofollow">History (20 revisions)</a> <a href="/nlab/show/created+limit/cite" style="color: black">Cite</a> <a href="/nlab/print/created+limit" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/created+limit" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>