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this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry is about the notion of “limit” in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. For the notion of the same name in <a class="existingWikiWord" href="/nlab/show/analysis">analysis</a> and <a class="existingWikiWord" href="/nlab/show/topology">topology</a> see <em><a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit of a sequence</a></em> and <em><a class="existingWikiWord" href="/nlab/show/limit+of+a+function">limit of a function</a></em>.</p> </blockquote> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" 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="limits_and_colimits">Limits and colimits</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/limit">limits and colimits</a></strong></p> <h2 id="1categorical">1-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit and colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">commutativity of limits and colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+limit">small limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+limit">connected limit</a>, <a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved limit</a>, <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected limit</a>, <a class="existingWikiWord" href="/nlab/show/created+limit">created limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product">product</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>, <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>, <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>, <a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, <a class="existingWikiWord" href="/nlab/show/direct+product">direct product</a>, <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></li> </ul> </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 and coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibered+limit">fibered limit</a></p> </li> </ul> <h2 id="2categorical">2-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isoinserter">isoinserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PIE-limit">PIE-limit</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> </ul> <h2 id="1categorical_2">(∞,1)-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/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/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+product">homotopy product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equalizer">homotopy equalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+totalization">homotopy totalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+end">homotopy end</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coproduct">homotopy coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+realization">homotopy realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#global_versus_local'>Global versus local</a></li> </ul> <li><a href='#terminology_and_notation'>Terminology and notation</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#local_definition_in_terms_of_representable_functors'>Local definition in terms of representable functors</a></li> <ul> <li><a href='#limit_of_a_setvalued_functor'>Limit of a Set-valued functor</a></li> <li><a href='#limit_of_a_functor_with_values_in_an_arbitrary_category'>Limit of a functor with values in an arbitrary category</a></li> <li><a href='#generalization_to_weighted_limits'>Generalization to weighted limits</a></li> <li><a href='#relation_to_continuous_functors'>Relation to continuous functors</a></li> </ul> <li><a href='#definition_in_terms_of_universal_cones'>Definition in terms of universal cones</a></li> <ul> <li><a href='#unwrapping'>Unwrapping</a></li> </ul> <li><a href='#global_definition_in_terms_of_adjoint_of_the_constant_diagram_functor'>Global definition in terms of adjoint of the constant diagram functor</a></li> <li><a href='#generalizations'>Generalizations</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#limits_in_analysis'>Limits in analysis</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#ConstructionFromProductsAndEqualizers'>Existence: construction from products and equalizers</a></li> <li><a href='#interaction_with_functor'>Interaction with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi></mrow><annotation encoding="application/x-tex">Hom</annotation></semantics></math>-functor</a></li> <li><a href='#in_'>In <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></a></li> <ul> <li><a href='#limits_in_set_are_homsets'>Limits in Set are hom-sets</a></li> </ul> <li><a href='#in_functor_categories'>In functor categories</a></li> <li><a href='#CompatibilityAmongUniversalConstructions'>Compatibility with universal constructions</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>In <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> a limit of a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D \to C</annotation></semantics></math> in 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>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">lim F</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> equipped with morphisms to the objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(d)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">d \in D</annotation></semantics></math>, such that everything in sight commutes. Moreover, the limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">lim F</annotation></semantics></math> is the <em>universal</em> object with this property, i.e. the “most optimized solution” to the problem of finding such an object.</p> <p>The limit construction has a wealth of applications throughout category theory and mathematics in general. In practice, it is possibly best thought of in the context of <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functors</a> as a <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> for maps into a diagram. So in some sense the limit object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">lim F</annotation></semantics></math> “subsumes” the entire diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(D)</annotation></semantics></math> into a single object, as far as morphisms <em>into</em> it are concerned. The corresponding universal object for morphisms <em>out</em> of the diagram is the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>.</p> <p>An intuitive general idea is that a limit of a diagram is the locus or solution set of a bunch of equations, where each of the coordinates is parametrized by one of the objects of the diagram, and where the equations are prescribed by the morphisms of the diagram. This idea is explained more formally <a href="http://ncatlab.org/nlab/show/limits+and+colimits+by+example#limitsintermsofotherops">here</a>.</p> <p>Often, the general theory of limits (but not colimits!) works better if the source of <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> is taken to be the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">D^op</annotation></semantics></math> (or equivalently, if <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> is taken to be a <a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant functor</a>). This is what we do below. In any given situation, of course, you use whatever categories and functors you're interested in.</p> <p>In some cases the category-theoretic notion of limit does reproduce notions of limit as known from analysis. See the examples below.</p> <h3 id="global_versus_local">Global versus local</h3> <p>In correspondence to the <em>local</em> definition of <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functors</a> (as discussed there), there is a <em>local</em> definition of limits (in terms of cones), that defines a limit (if it exists) for each individual diagram, and there is a <em>global</em> definition, which defines the limit for <em>all</em> diagrams (in terms of an <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint</a>).</p> <p>If all limits over the given shape of diagrams exist in a category, then both definitions are equivalent.</p> <p>See also the analogous discussion at <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>.</p> <h2 id="terminology_and_notation">Terminology and notation</h2> <p>A limit is taken over 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>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D^{op} \to C</annotation></semantics></math> and since the functor comes equipped with the information about what its domain is, one can just write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\lim F</annotation></semantics></math> for its limit. But often it is helpful to indicate how the functor is evaluated on objects, in which case the limit is written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lim_{d \in D} F(d)</annotation></semantics></math>; this is used particularly when <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> is given by a formula (as with other notation with bound variables.)</p> <p>In some schools of mathematics, limits are called <em>projective limits</em>, while colimits are called <em>inductive limits</em>. Also seen are (respectively) <em>inverse limits</em> and <em>direct limits</em>. Both these systems of terminology are alternatives to using ‘co-’ when distinguishing limits and colimits. The first system also appears in <a class="existingWikiWord" href="/nlab/show/pro-object">pro-object</a> and <a class="existingWikiWord" href="/nlab/show/ind-object">ind-object</a>.</p> <p>Correspondingly, the symbols <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>←</mo></munder></mrow><annotation encoding="application/x-tex">\underset{\leftarrow}lim</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mo>→</mo></munder></mrow><annotation encoding="application/x-tex">\underset{\rightarrow}\lim</annotation></semantics></math> are used instead of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi></mrow><annotation encoding="application/x-tex">\lim</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">colim</mo></mrow><annotation encoding="application/x-tex">\colim</annotation></semantics></math>.</p> <p>Confusingly, many authors restrict the meanings of these alternative terms to (co)limits whose sources are <a class="existingWikiWord" href="/nlab/show/direction">directed sets</a>; see <a class="existingWikiWord" href="/nlab/show/directed+limit">directed limit</a>. In fact, this is the original meaning; projective and inductive limits in this sense were studied in algebra before the general category-theoretic notion of (co)limit.</p> <h2 id="definition">Definition</h2> <h3 id="local_definition_in_terms_of_representable_functors">Local definition in terms of representable functors</h3> <p>There is a general abstract definition of limits in terms of representable functors, which we describe now. This reproduces the more concrete and maybe more familiar description in terms of universal cones, which is described further below.</p> <p>Let in the following <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and <a class="existingWikiWord" href="/nlab/show/Set">Set</a> the category of sets (possibly realized as the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mi>Set</mi></mrow><annotation encoding="application/x-tex">U Set</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-small sets with respect to a given <a class="existingWikiWord" href="/nlab/show/Grothendieck+universe">Grothendieck universe</a>.)</p> <h4 id="limit_of_a_setvalued_functor">Limit of a Set-valued functor</h4> <p>The <strong>limit of a Set-valued functor</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F : D^{op} \to Set</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi><mo>≔</mo><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> lim F \coloneqq Hom_{[D^{op}, Set]}(pt, F) \in Set </annotation></semantics></math></div> <p>in 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><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D^{op}, Set]</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> category), where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> pt : D^{op} \to Set </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo>:</mo><mi>d</mi><mo>↦</mo><mo stretchy="false">{</mo><mo>*</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> pt : d \mapsto \{*\} </annotation></semantics></math></div> <p>is the functor constant on the <a class="existingWikiWord" href="/nlab/show/point">point</a>, i.e. the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> diagram.</p> <p>The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">lim F</annotation></semantics></math> is equivalently called</p> <ul> <li> <p>the set of <a class="existingWikiWord" href="/nlab/show/global+section">global sections</a> of <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>;</p> </li> <li> <p>the set of <a class="existingWikiWord" href="/nlab/show/global+element">global elements</a> of <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>.</p> </li> </ul> <p>The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">lim F</annotation></semantics></math> can be equivalently expressed as an <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> of a <a class="existingWikiWord" href="/nlab/show/product">product</a>, explicitly:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi><mo>≃</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>d</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mo>∈</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>∀</mo><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>i</mi></msub><mover><mo>→</mo><mi>α</mi></mover><msub><mi>d</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>D</mi><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mrow><msub><mi>d</mi> <mi>j</mi></msub></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi> <mrow><msub><mi>d</mi> <mi>i</mi></msub></mrow></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> lim F \simeq \left\lbrace (x_d)_{d \in D} \in \prod_{d \in D} F(d) | \forall (d_i \stackrel{\alpha}{\to} d_j) \in D : F(\alpha)(x_{d_j}) = x_{d_i} \right\rbrace </annotation></semantics></math></div> <p>In particular, the limit of a set-valued functor always exists.</p> <p>Notice the important triviality that the covariant <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> commutes with set-valued limits: for every set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> we have a bijection of sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>lim</mi><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Hom_{Set}(S, lim F) \simeq \lim Hom_{Set}(S, F(-)) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Hom(S, F(-)) : D^{op} \to Set</annotation></semantics></math>.</p> <h4 id="limit_of_a_functor_with_values_in_an_arbitrary_category">Limit of a functor with values in an arbitrary category</h4> <p>The above formula generalizes straightforwardly to a notion of limit for functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D^{op} \to C</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> an arbitrary category if we construct a certain <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> which we will call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>lim</mi><mo stretchy="false">^</mo></mover><mi>F</mi></mrow><annotation encoding="application/x-tex">\hat \lim F</annotation></semantics></math>. The actual limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">lim F</annotation></semantics></math> is then, if it exists, the object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/representable+functor">representing</a> this presheaf.</p> <p>More precisely, using the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>:</mo><mi>C</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">y: C \to [C^{op}, Set]</annotation></semantics></math> define for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D^{op} \to C</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>lim</mi><mo stretchy="false">^</mo></mover><mi>F</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\hat \lim F \in [C^{op}, Set]</annotation></semantics></math> by</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><mover><mi>lim</mi><mo stretchy="false">^</mo></mover><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>Hom</mi> <mrow><msup><mi>Set</mi> <mrow><msup><mi>D</mi> <mi>op</mi></msup></mrow></msup></mrow></msub><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\hat \lim F)(c)\coloneqq Hom_{Set^{D^{op}}}(pt,Hom_C(c,F(-))) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math>, or suppressing the subscripts for readability:</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><mover><mi>lim</mi><mo stretchy="false">^</mo></mover><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\hat lim F)(c) = Hom(pt , Hom(c,F(-))) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a>-valued limit always exists; iff this presheaf is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> by an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\lim F</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, then this is the <strong>limit</strong> of <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom(c, \lim F) \simeq Hom(pt, Hom(c,F(-))) \,. </annotation></semantics></math></div> <h4 id="generalization_to_weighted_limits">Generalization to weighted limits</h4> <p>In the above formulation, there is an evident generalization to <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limits</a>:</p> <p>replace in the above the constant terminal functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">pt : D^{op} \to Set</annotation></semantics></math> with <em>any</em> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">W : D^{op} \to Set</annotation></semantics></math> – then called the <em>weight</em> –, then the <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>-weighted limit of <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mi>W</mi></munder><mi>F</mi></mrow><annotation encoding="application/x-tex"> \lim_W F </annotation></semantics></math></div> <p>often written</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>W</mi><mo>,</mo><mi>F</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \{W,F\} </annotation></semantics></math></div> <p>is, if it exists, the object representing the presheaf</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>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> c \mapsto Hom_{[D^{op}, Set]}(W , Hom_C(c,F(-))) \,, </annotation></semantics></math></div> <p>i.e. such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><munder><mi>lim</mi> <mi>W</mi></munder><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex"> Hom(c, \lim_W F) \simeq Hom(W, Hom(c,F(-))) \, </annotation></semantics></math></div> <p>naturally in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math>.</p> <h4 id="relation_to_continuous_functors">Relation to continuous functors</h4> <p>The very definition of limit as above asserts that the covariant <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Hom(c,-) : C \to Set</annotation></semantics></math> commutes with forming limits. Indeed, the definition is equivalent to saying that the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> is a <a class="existingWikiWord" href="/nlab/show/continuous+functor">continuous functor</a>.</p> <h3 id="definition_in_terms_of_universal_cones">Definition in terms of universal cones</h3> <p>Unwrapping the above abstract definition of limits yields the following more hands-on description in terms of universal cones.</p> <h4 id="unwrapping">Unwrapping</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D^{op} \to C</annotation></semantics></math> be a functor.</p> <p>Notice that for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> an element</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> * \to Hom(pt, Hom(c, F(-))) </annotation></semantics></math></div> <p>is to be identified with a collection of morphisms</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><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> c \to F(d) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">d \in D</annotation></semantics></math>, such that all triangles</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && c \\ & \swarrow && \searrow \\ F(d_i) && \stackrel{F(f)}{\to} && F(d_j) } </annotation></semantics></math></div> <p>commute. Such a collection of morphisms is called a <strong>cone</strong> over <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>, for the obvious reason.</p> <p>If the limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\lim F \in C</annotation></semantics></math> of <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> exist, then it singles out a special cone given by the composite morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mover><mo>→</mo><mrow><mo>*</mo><mo>↦</mo><msub><mi>Id</mi> <mrow><mi>lim</mi><mi>F</mi></mrow></msub></mrow></mover><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>lim</mi><mi>F</mi><mo>,</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>Hom</mi><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>lim</mi><mi>F</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> * \stackrel{* \mapsto Id_{\lim F}}{\to} Hom_C(\lim F, \lim F) \stackrel{\simeq}{\to} Hom(pt, Hom(\lim F, F(-))) \,, </annotation></semantics></math></div> <p>where the first morphism picks the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\lim F</annotation></semantics></math> and the second one is the defining bijection of a limit as above.</p> <p>The cone</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>lim</mi><mi>F</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \lim F \\ & \swarrow && \searrow \\ F(d_i) && \stackrel{F(f)}{\to} && F(d_j) } </annotation></semantics></math></div> <p>is called the <strong>universal cone</strong> over <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>, because, again by the defining property of limit as above, every cone <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>c</mi><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{c \to F(d)\}_{d \in D}</annotation></semantics></math> as above is bijectively related to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>→</mo><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">c \to \lim F</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mover><mo>→</mo><mrow><mo stretchy="false">{</mo><mi>c</mi><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub></mrow></mover><mi>Hom</mi><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> * \stackrel{\{c \to F(d)\}_{d \in D}}{\to} Hom(pt, Hom(c, F(-))) \stackrel{\simeq}{\to} Hom(c, \lim F) \,. </annotation></semantics></math></div> <p>By inspection one finds that, indeed, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>→</mo><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">c \to \lim F</annotation></semantics></math> is the morphism which exhibits the factorization of the cone <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>c</mi><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{c \to F(d)\}_{d \in D}</annotation></semantics></math> through the universal limit cone</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>=</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>lim</mi><mi>F</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && c \\ & \swarrow && \searrow \\ F(d_i) && \stackrel{F(f)}{\to} && F(d_j) } = \array{ && c \\ && \downarrow \\ && \lim F \\ & \swarrow && \searrow \\ F(d_i) && \stackrel{F(f)}{\to} && F(d_j) } \,. </annotation></semantics></math></div> <p>An illustrative example is the following: a limit of the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Id</mi> <mi>c</mi></msub><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Id_c:C\to C</annotation></semantics></math> is, if it exists, an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <h3 id="global_definition_in_terms_of_adjoint_of_the_constant_diagram_functor">Global definition in terms of adjoint of the constant diagram functor</h3> <p>Given categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> and <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>, limits over functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">D^{op} \to C</annotation></semantics></math> may exist for some functors, but not for all. If it does exist for all functors, then the above <em>local definition</em> of limits is equivalent to the following <em>global definition</em>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> any category, 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><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D^{op},C]</annotation></semantics></math> is the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/diagram">diagrams</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Pullback along the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex">D^{op} \to pt</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi><mo>=</mo><mo stretchy="false">{</mo><mo>•</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">pt = \{\bullet\}</annotation></semantics></math> induces a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>const</mi><mo>:</mo><mi>C</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> const : C \to [D^{op},C] </annotation></semantics></math></div> <p>which sends every object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to the diagram functor constant on this object.</p> <p>The <a class="existingWikiWord" href="/nlab/show/adjoint+functor">left adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>colim</mi> <mi>D</mi></msub><mo>:</mo><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> colim_D : [D^{op},C] \to C </annotation></semantics></math></div> <p>of this functor is, if it exists, the functor which sends every diagram to its <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> and the <a class="existingWikiWord" href="/nlab/show/adjoint+functor">right adjoint</a> is, if it exists, the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mi>D</mi></msub><mo>:</mo><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> lim_D : [D^{op},C] \to C </annotation></semantics></math></div> <p>which sends every diagram to its <a class="existingWikiWord" href="/nlab/show/limit">limit</a>. The Hom-isomorphisms of these <a class="existingWikiWord" href="/nlab/show/adjunction">adjunctions</a> state precisely the universal property of <a class="existingWikiWord" href="/nlab/show/limit">limit</a> and <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> given above.</p> <p>Concretely this means that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> we have a bijection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>const</mi> <mi>c</mi></msub><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_C(c, \lim F) \simeq Hom_{[D^{op},C]}(const_c, F) \,. </annotation></semantics></math></div> <p>From this perspective, a limit is a special case of a <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a>, as described there, namely a Kan extension to the <a class="existingWikiWord" href="/nlab/show/point">point</a>.</p> <h3 id="generalizations">Generalizations</h3> <p>The notion of limit, being fundamental to <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, generalizes to many other situations. Examples include the following.</p> <ul> <li>In <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> one has the notion of <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a>. This makes sense in ordinary category theory as well, but it turns out that in that case all weighted limits reduce to ordinary “conical” ones.</li> <li>Weighted limits generalize further to the notion of limit in a <a class="existingWikiWord" href="/nlab/show/2-category+equipped+with+proarrows">2-category equipped with proarrows</a>, or to a <a class="existingWikiWord" href="/nlab/show/Yoneda+structure">Yoneda structure</a>.</li> <li>In <a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a> one has the notion of a <a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a>.</li> <li>Similarly, in <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-category</a> theory there is a notion of a limit. Using quasicategories as a model for <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, the definition is a straightforward extension of the one as a terminal cone: it is simply a <a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">quasi-categorical terminal object</a> in the quasicategory of cones. See <a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit in quasi-categories</a>.</li> <li>One expects that similarly, all sorts of <a class="existingWikiWord" href="/nlab/show/higher+categories">higher categories</a> have their own appropriate notions of limit and colimit.</li> </ul> <h2 id="examples">Examples</h2> <p>The central point about examples of limits is:</p> <p><em>Categorical limits are ubiquitous</em>.</p> <p>To a fair extent, <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> is all about limits and the other <a class="existingWikiWord" href="/nlab/show/universal+construction">universal constructions</a>: <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extensions</a>, <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functors</a>, <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functors</a>, which are all special cases of limits – and limits are special cases of these.</p> <p>Listing examples of limits in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> is much like listing examples of <a class="existingWikiWord" href="/nlab/show/integral">integrals</a> in <a class="existingWikiWord" href="/nlab/show/analysis">analysis</a>: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see <a class="existingWikiWord" href="/nlab/show/coend">coend</a> for more).</p> <p>Keeping that in mind, we do list some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.</p> <h3 id="general">General</h3> <ul> <li>For a pedagogical list of examples see <a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a>.</li> </ul> <p>Here are some important examples of limits, classified by the shape of the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>:</p> <ul> <li>A limit of the <a class="existingWikiWord" href="/nlab/show/empty+set">empty diagram</a> is a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>.</li> <li>A limit of a diagram consisting of two (or more) objects and no nontrivial morphisms is their <a class="existingWikiWord" href="/nlab/show/product">product</a>.</li> <li>A limit of a <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a> is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>.</li> <li>A limit of a pair (or more) of <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> is an <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>.</li> <li>A limit over a <a class="existingWikiWord" href="/nlab/show/finite+category">finite category</a> is a <a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a>.</li> <li>Another important “shape” of limits are those that give rise to <a class="existingWikiWord" href="/nlab/show/end">ends</a>.</li> <li>A limit of a diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D \to C</annotation></semantics></math> for which <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> admits an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>, is simply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(I)</annotation></semantics></math>.</li> </ul> <h3 id="limits_in_analysis">Limits in analysis</h3> <p>The concept of <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit of a sequence</a> in <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> is a special case of category theoretic limits, see <a href="limit+of+a+sequence#RelationToLimitsInCategoryTheory">there</a>.</p> <h2 id="properties">Properties</h2> <h3 id="ConstructionFromProductsAndEqualizers">Existence: construction from products and equalizers</h3> <p>Frequently some limits can be computed in terms of other limits. This makes things easier since we only have to assume that categories have, or functors preserve, some easier-to-verify class of limits in order to obtain results about a larger one.</p> <p>The most common example of this is the computation of limits in terms of products and equalizers. Specifically, if the limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D^{op} \to C</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/products">products</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{d\in Obj(D)} F(d)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mi>d</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{f\in Mor{d}} F(s(f))</annotation></semantics></math> all exist, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">lim F</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/subobject">subobject</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{d\in Obj(D)} F(d)</annotation></semantics></math>, namely the <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mrow><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \prod_{d \in Obj(D)} F(d) \stackrel{\prod_{f \in Mor(d)} (F(f) \circ p_{s(f)}) }{\to} \prod_{f \in Mor(D)} F(s(f)) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>d</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>t</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \prod_{d \in Obj(D)} F(d) \stackrel{\prod_{f \in Mor(d)} (p_{t(f)}) }{\to} \prod_{f \in Mor(D)} F(s(f)) \,. </annotation></semantics></math></div> <p>Conversely, if both of these products exist and so does the equalizer of this pair of maps, then that equalizer is a limit of <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>. In particular, therefore, a category has all limits as soon as it has all products and equalizers, and a functor defined on such a category <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> all limits as soon as it preserves products and equalizers.</p> <p>(More precisely, it suffices only to consider equalizers of <a class="existingWikiWord" href="/nlab/show/reflexive+pairs">reflexive pairs</a>.)</p> <p>Another example is that all <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> can be computed in terms of <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> and a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>.</p> <h3 id="interaction_with_functor">Interaction with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi></mrow><annotation encoding="application/x-tex">Hom</annotation></semantics></math>-functor</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/hom-functor+preserves+limits">hom-functor preserves limits</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+small">locally small</a> category, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D^{op} \to C</annotation></semantics></math> a functor and writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">C(c, F(-)) : D^{op} \to Set</annotation></semantics></math>, we have</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 stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>lim</mi><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C(c, lim F) \simeq lim C(c, F(-)) \,. </annotation></semantics></math></div></div> <p>Depending on how one introduces limits this holds by definition or is an easy consequence.</p> <h3 id="in_">In <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></h3> <div class="num_prop"> <h6 id="limits_in_set_are_homsets">Limits in Set are hom-sets</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F : D^{op} \to Set</annotation></semantics></math> any functor and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mo>*</mo></msub><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">const_{*} : D^{op} \to Set</annotation></semantics></math> the functor constant on the <a class="existingWikiWord" href="/nlab/show/point">point</a>, the limit of <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> is the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi><mo>≃</mo><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msub><mi>const</mi> <mo>*</mo></msub><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> lim F \simeq [D^{op}, Set](const_{*}, F) </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a>, i.e. the set of <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformations</a> from the constant functor into <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>.</p> </div> <h3 id="in_functor_categories">In functor categories</h3> <div class="un_prop"> <h6 id="proposition_2">Proposition</h6> <p><strong>(limits in functor categories are computed pointwise)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be a small category and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">D'</annotation></semantics></math> be any category. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a category which admits limits of shape <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>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>′</mo><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D',C]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a>. Then</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>′</mo><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D',C]</annotation></semantics></math> admits <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>-shaped limits;</li> <li>these limits are computed objectwise (“pointwise”) in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy="false">[</mo><mi>D</mi><mo>′</mo><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F : D^{op} \to [D',C]</annotation></semantics></math> a functor we have for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>′</mo><mo>∈</mo><mi>D</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">d' \in D'</annotation></semantics></math> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>lim</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(lim F)(d') \simeq lim (F(-)(d'))</annotation></semantics></math>. Here the limit on the right is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</li> </ul> </div> <h3 id="CompatibilityAmongUniversalConstructions">Compatibility with universal constructions</h3> <div class="num_prop" id="RightAdjointsPreserveLimits"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/right+adjoints+preserve+limits">right adjoints preserve limits</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">R \;\colon\; C \to C'</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> that is <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> to some functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mi>C</mi><mo>′</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">L : C' \to C</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> 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> admits <a class="existingWikiWord" href="/nlab/show/limits">limits</a> of shape <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>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/preserved+limit">commutes with</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-shaped limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in that</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D^{op} \to C</annotation></semantics></math> some diagram, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>lim</mi><mo stretchy="false">(</mo><mi>R</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R(lim F) \simeq lim (R \circ F) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Using the adjunction isomorphism and the above fact that <a class="existingWikiWord" href="/nlab/show/hom-functor+preserves+limits">hom-functor preserves limits</a>, one obtains for every <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><mo>′</mo></mrow><annotation encoding="application/x-tex">c' \in C'</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mi>C</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>lim</mi><mi>C</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>lim</mi><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>R</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>lim</mi><mo stretchy="false">(</mo><mi>R</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} C'(c', R (lim F)) & \simeq C(L(c'), lim F) \\ & \simeq lim C(L(c'), F) \\ & \simeq lim C'(c', R\circ F) \\ & \simeq C'(c', lim (R \circ F)) \,. \end{aligned} \,. </annotation></semantics></math></div> <p>Since this holds naturally for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">c'</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma, corollary II</a> on uniqueness of representing objects implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>lim</mi><mo stretchy="false">(</mo><mi>R</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R (lim F) \simeq lim (R \circ F)</annotation></semantics></math>.</p> </div> <div class="num_prop" id="LimitsCommuteWithLimits"> <h6 id="proposition_4">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/limits+commute+with+limits">limits commute with limits</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">D'</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/small+categories">small categories</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a category which admits <a class="existingWikiWord" href="/nlab/show/limits">limits</a> of shape <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> as well as <a class="existingWikiWord" href="/nlab/show/limits">limits</a> of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">D'</annotation></semantics></math>. Then these limits commute with each other, in that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>×</mo><msup><mrow><mi>D</mi><mo>′</mo></mrow> <mi>op</mi></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D^{op} \times {D'}^{op} \to C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> , with corresponding induced functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>D</mi></msub><mo>:</mo><msup><mrow><mi>D</mi><mo>′</mo></mrow> <mi>op</mi></msup><mo>→</mo><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F_D : {D'}^{op} \to [D^{op},C]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><mi>D</mi><mo>′</mo></mrow></msub><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy="false">[</mo><msup><mrow><mi>D</mi><mo>′</mo></mrow> <mi>op</mi></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F_{D'} : {D}^{op} \to [{D'}^{op},C]</annotation></semantics></math>, then the canonical comparison morphism</p> <div class="maruku-equation" id="eq:ComparisonMorphismForCommutingLimits"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi><mo>≃</mo><msub><mi>lim</mi> <mi>D</mi></msub><mo stretchy="false">(</mo><msub><mi>lim</mi> <mrow><mi>D</mi><mo>′</mo></mrow></msub><msub><mi>F</mi> <mi>D</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>lim</mi> <mrow><mi>D</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>lim</mi> <mi>D</mi></msub><msub><mi>F</mi> <mrow><mi>D</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> lim F \simeq lim_{D} (lim_{D'} F_D ) \simeq lim_{D'} (lim_{D} F_{D'} ) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Since the <a class="existingWikiWord" href="/nlab/show/limit">limit</a>-construction is the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> functor to the <a class="existingWikiWord" href="/nlab/show/constant+functor">constant</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>-functor, this is a special case of <em><a class="existingWikiWord" href="/nlab/show/right+adjoints+preserve+limits">right adjoints preserve limits</a></em> (Prop. <a class="maruku-ref" href="#RightAdjointsPreserveLimits"></a>).</p> </div> <p>See <a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a> for what formula <a class="maruku-eqref" href="#eq:ComparisonMorphismForCommutingLimits">(1)</a> says for instance for the special case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p><strong>(general non-commutativity of limits with colimits)</strong></p> <p>In general limits do <em>not</em> commute with <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. But under a number of special conditions of interest they do. Special cases and concrete examples are discussed at <em><a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">commutativity of limits and colimits</a></em>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>limit</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+limit">internal limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>, <a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a>, <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a>, <a class="existingWikiWord" href="/nlab/show/end">end</a>, <a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> </ul> </li> </ul> <h2 id="references">References</h2> <p>Limits and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> were defined in <a class="existingWikiWord" href="/nlab/show/Daniel+M.+Kan">Daniel M. Kan</a> in Chapter II of the paper that also defined <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> and <a class="existingWikiWord" href="/nlab/show/Kan+extensions">Kan extensions</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+M.+Kan">Daniel M. Kan</a>, <em>Adjoint functors</em>, Transactions of the American Mathematical Society 87:2 (1958), 294–294 (<a href="https://doi.org/10.1090/s0002-9947-1958-0131451-0">doi:10.1090/s0002-9947-1958-0131451-0</a>)</li> </ul> <p>This paper refers to limits as <em>inverse limits</em>.</p> <p>The observation that limits can be constructed from <a class="existingWikiWord" href="/nlab/show/products">products</a> and <a class="existingWikiWord" href="/nlab/show/equalisers">equalisers</a> is due to:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/J.-M.+Maranda">J.-M. Maranda</a>, <em>Some remarks on limits in categories</em>. Canadian Mathematical Bulletin <strong>5</strong> 2 (1962) 133-146 [<a href="https://doi.org/10.4153/CMB-1962-015-0">doi:10.4153/CMB-1962-015-0</a>]</li> </ul> <p>That, more generally, it suffices to consider only equalisers of <a class="existingWikiWord" href="/nlab/show/reflexive+pairs">reflexive pairs</a> is due to:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ernest+Manes">Ernest Manes</a>, <em>A triple miscellany: Some aspects of the theory of algebra over a triple</em>, PhD thesis, Wesleyan University, 1967. [<a href="https://github.com/CategoryTheoryArchive/archive/blob/main/resources/1967_manes_triple-miscellany.pdf">pdf</a>]</p> </li> <li> <p>S. A. Huq and R. Cornu. <em>A remark on the existence of limits in categories</em>. Mathematische Nachrichten 55.1‐6 (1973): 223-224. [ <a href="https://doi.org/10.1002/mana.19730550111">doi:10.1002/mana.19730550111</a>]</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, §III.4 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 28, 2024 at 08:20:15. 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