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name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #116 to #117: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='category_theory'>Category theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></strong></p> <h2 id='sidebar_concepts'>Concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a></p> </li> </ul> <h2 id='sidebar_universal_constructions'>Universal constructions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal construction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>/<a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a>/<a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+lifting+theorem'>adjoint lifting theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gabriel%E2%80%93Ulmer+duality'>Gabriel-Ulmer duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freyd-Mitchell+embedding+theorem'>Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> </ul> <h2 id='sidebar_applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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> <h4 id='notions_of_subcategory'>Notions of subcategory</h4> <div class='hide'> <ul> <li> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/subcategory'>subcategory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full</a>, <a class='existingWikiWord' href='/nlab/show/diff/wide+subcategory'>wide</a>, <a class='existingWikiWord' href='/nlab/show/diff/dense+subcategory'>dense</a>, <a class='existingWikiWord' href='/nlab/show/diff/replete+subcategory'>replete</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflective</a>, <a class='existingWikiWord' href='/nlab/show/diff/coreflective+subcategory'>coreflective</a>, <a class='existingWikiWord' href='/nlab/show/diff/bireflective+subcategory'>bireflective</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+subcategory'>abelian</a>, <a class='existingWikiWord' href='/nlab/show/diff/Serre+subcategory'>Serre</a>, <a class='existingWikiWord' href='/nlab/show/diff/topologizing+subcategory'>topologizing</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/internal+subcategory'>internal subcategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dense+sub-site'>dense sub-site</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/full+sub-2-category'>sub-2-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/full+sub-2-category'>full sub-2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+full+sub-2-category'>locally full sub-2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-2-category'>reflective sub-2-category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localizing+subcategory'>localizing subcategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sub-%28infinity%2C1%29-category'>sub-(∞,1)-category</a> (<a class='existingWikiWord' href='/nlab/show/diff/sub-%28infinity%2C1%29-category+-+internal+formulation'>internal formulation</a>)</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-category</a> (<a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category+-+internal+formulation'>internal formulation</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+product-preserving+sub-%28%E2%88%9E%2C1%29-category'>reflective product-preserving sub-(∞,1)-category</a> (<a class='existingWikiWord' href='/nlab/show/diff/reflective+product-preserving+sub-%28%E2%88%9E%2C1%29-category+-+internal+formulation'>internal formulation</a>)</p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/thick+subcategory'>thick subcategory</a> (of a <a class='existingWikiWord' href='/nlab/show/diff/triangulated+category'>triangulated category</a>)</p> </li> </ul> <div> <p> <a href='/nlab/edit/notions+of+subcategory'>Edit this sidebar</a> </p> </div></div> <h4 id='modalities_closure_and_reflection'>Modalities, Closure and Reflection</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/modal+type+theory'>modal type theory</a></strong>, <a class='existingWikiWord' href='/nlab/show/diff/modal+logic'>modal logic</a></p> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/closure+operator'>closure operator</a></strong>, <a class='existingWikiWord' href='/nlab/show/diff/universal+closure+operator'>universal closure operator</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/idempotent+monad'>idempotent monad</a>, <a class='existingWikiWord' href='/nlab/show/diff/comonad'>comonad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/modal+type'>modal type</a>, <a class='existingWikiWord' href='/nlab/show/diff/local+object'>local object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflective subcategory</a>, <a class='existingWikiWord' href='/nlab/show/diff/coreflective+subcategory'>coreflective subcategory</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Moore+closure'>Moore closure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/modal+type+theory'>geometric modality</a>/<a class='existingWikiWord' href='/nlab/show/diff/Lawvere-Tierney+topology'>Lawvere-Tierney topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/S4+modal+logic'>S4 modal logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-truncated+object+of+an+%28infinity%2C1%29-category'>n-truncation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/flat+modality'>flat modality</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/sharp+modality'>sharp modality</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#Idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#characterizations'>Characterizations</a></li><li><a href='#special_cases'>Special cases</a><ul><li><a href='#ExactReflectiveSubcategories'>Exact reflective subcategories</a></li><li><a href='#complete_reflective_subcategories'>Complete reflective subcategories</a></li><li><a href='#AccessibleReflectiveSubcategories'>Accessible reflective subcategories</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#general'>General</a></li><li><a href='#AsEilenbergMooreCategory'>As Eilenberg-Moore category of the idempotent monad</a></li><li><a href='#reflective_subcategories_of_locally_presentable_categories'>Reflective subcategories of locally presentable categories</a></li><li><a href='#ReflectiveSubcategoriesOfCartesianClosedCategotries'>Reflective subcategories of cartesian closed categories</a></li><li><a href='#reflective_and_coreflective_subcategories'>Reflective and coreflective subcategories</a></li><li><a href='#property_vs_structure'>Property vs structure</a></li></ul></li><li><a href='#Examples'>Examples</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='Idea'>Idea</h2> <p>A <em>reflective subcategory</em> is a <a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full subcategory</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>↪</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'> C \hookrightarrow D </annotation></semantics></math></div> <p>such that <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math> and <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>d</mi><mo>→</mo><mi>d</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>f \colon d \to d'</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> have “reflections” <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>d</mi></mrow><annotation encoding='application/x-tex'>T d</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>T</mi><mi>d</mi><mo>→</mo><mi>T</mi><mi>d</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>T f \colon T d \to T d'</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. Every object in <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> looks at its own reflection via a morphism <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>→</mo><mi>T</mi><mi>d</mi></mrow><annotation encoding='application/x-tex'>d \to T d</annotation></semantics></math> and the reflection of an object <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> is equipped with an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>c</mi><mo>≅</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>T c \cong c</annotation></semantics></math>.</p> <p>A canonical example is the inclusion</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi mathvariant='normal'>Ab</mi><mo>↪</mo><mi mathvariant='normal'>Grp</mi></mrow><annotation encoding='application/x-tex'> \mathrm{Ab} \hookrightarrow \mathrm{Grp} </annotation></semantics></math></div> <p>of the <a class='existingWikiWord' href='/nlab/show/diff/Ab'>category of abelian groups</a> into the <a class='existingWikiWord' href='/nlab/show/diff/Grp'>category of groups</a>, whose reflector is the operation of <em><a class='existingWikiWord' href='/nlab/show/diff/abelianization'>abelianization</a></em>.</p> <p>A useful property of reflective subcategories is that the inclusion <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>↪</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>C \hookrightarrow D</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/created+limit'>creates all limits</a> of <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has all <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a> which <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> admits.</p> <h2 id='definition'>Definition</h2> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>A <a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full subcategory</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>C</mi><mo>↪</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>i : C \hookrightarrow D </annotation></semantics></math> is <strong>reflective</strong> if the inclusion <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>T</mi><mo>⊣</mo><mi>i</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>C</mi><mover><mo>↪</mo><mover><mo>←</mo><mi>T</mi></mover></mover><mi>D</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (T \dashv i) : C \stackrel{\stackrel{T}{\leftarrow}}{\hookrightarrow} D \,. </annotation></semantics></math></div></div> <p>The left adjoint is sometimes called the <strong>reflector</strong>, and a functor which is a reflector (or has a fully faithful right adjoint, which is the same up to equivalence) is called a <strong>reflection</strong>. Of course, there are dual notions of <strong><a class='existingWikiWord' href='/nlab/show/diff/coreflective+subcategory'>coreflective subcategory</a></strong>, <strong>coreflector</strong>, and <strong>coreflection</strong>.</p> <div class='num_remark' id='NonFullReflections'> <h6 id='remark'>Remark</h6> <p>A few sources (such as <a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>MacLane 1971</a>) do not require a reflective subcategory to be full. However, in light of the fact that non-full subcategories are not <a class='existingWikiWord' href='/nlab/show/diff/principle+of+equivalence'>invariant under equivalence</a>, consideration of non-full reflective subcategories seems of limited usefulness. The general consensus among category theorists nowadays seems to be that “reflective subcategory” implies fullness. Examples for non-full subcategories and their behaviour can be found in a <a href='http://www.tac.mta.ca/tac/volumes/30/41/30-41.pdf'>TAC paper</a> by Adámek and Rosický.</p> </div> <div class='num_remark'> <h6 id='remark_2'>Remark</h6> <p>The components of the <a class='existingWikiWord' href='/nlab/show/diff/unit+of+an+adjunction'>unit</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mi>η</mi></msup></mtd> <mtd><msup><mo>↘</mo> <mi>Id</mi></msup></mtd></mtr> <mtr><mtd><mi>D</mi></mtd> <mtd><mover><mo>→</mo><mi>T</mi></mover></mtd> <mtd><mi>C</mi></mtd> <mtd><mover><mo>↪</mo><mi>i</mi></mover></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ & \nearrow &\Downarrow^{\eta}& \searrow^{Id} \\ D &\stackrel{T}{\to}& C &\stackrel{i}{\hookrightarrow} & D } </annotation></semantics></math></div> <p>of this <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> “reflect” each object <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>d \in D</annotation></semantics></math> into its image <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>d</mi></mrow><annotation encoding='application/x-tex'>T d</annotation></semantics></math> in the reflective subcategory</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>η</mi> <mi>d</mi></msub><mo>:</mo><mi>d</mi><mo>→</mo><mi>T</mi><mi>d</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \eta_d : d \to T d \,. </annotation></semantics></math></div></div> <p>This reflection is sometimes called a <em><a class='existingWikiWord' href='/nlab/show/diff/localization'>localization</a></em> (due to <a href='reflective+localization#ReflectiveSubcategoriesAreLocalizations'>this Prop.</a> at <em><a class='existingWikiWord' href='/nlab/show/diff/reflective+localization'>reflective localization</a></em>), although sometimes this term is reserved for the case when the functor <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>left exact</a>.</p> <div class='num_defn'> <h6 id='definition_3'>Definition</h6> <p>If the reflector <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/faithful+functor'>faithful</a>, the reflection is called a <strong><a class='existingWikiWord' href='/nlab/show/diff/completion'>completion</a></strong>.</p> </div> <h2 id='characterizations'>Characterizations</h2> <div class='num_prop' id='CharacterizationByLocalization'> <h6 id='proposition'>Proposition</h6> <p><strong>(equivalent characterizations)</strong></p> <p>Given any pair of <a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functors</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>:</mo><mspace width='thickmathspace' /><mi>B</mi><munderover><mo>⊥</mo><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mi>L</mi></mover></munderover><mi>A</mi></mrow><annotation encoding='application/x-tex'> (L \dashv R) \;:\; B \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} A </annotation></semantics></math></div> <p>the following are equivalent:</p> <ol> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/full+and+faithful+functor'>fully faithful</a>. (In this case <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> is equivalent to its <a class='existingWikiWord' href='/nlab/show/diff/essential+image'>essential image</a> in <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> under <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, a full <a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflective subcategory</a> of <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.)</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/unit+of+an+adjunction'>counit</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ε</mi><mo>:</mo><mi>L</mi><mi>R</mi><mo>→</mo><msub><mn>1</mn> <mi>B</mi></msub></mrow><annotation encoding='application/x-tex'>\varepsilon : L R \to 1_B</annotation></semantics></math> of the <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> is a <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>natural isomorphism</a> of functors.</p> </li> <li> <p>There exists some natural isomorphism <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>R</mi><mo>→</mo><msub><mn>1</mn> <mi>B</mi></msub></mrow><annotation encoding='application/x-tex'>L R \to 1_B</annotation></semantics></math>.</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>R</mi><mi>L</mi><mo>,</mo><mi>R</mi><mi>ε</mi><mi>L</mi><mo>,</mo><mi>η</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(R L, R\varepsilon L,\eta)</annotation></semantics></math> associated with the adjunction is <a class='existingWikiWord' href='/nlab/show/diff/idempotent+monad'>idempotent</a>, the right adjoint <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/conservative+functor'>conservative</a>, and the left adjoint <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/essentially+surjective+functor'>essentially surjective on objects</a>.</p> </li> <li> <p>If <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is the set of morphisms <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>s</mi></mrow><annotation encoding='application/x-tex'>s</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo stretchy='false'>(</mo><mi>s</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>L(s)</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> in <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>L \colon A \to B</annotation></semantics></math> realizes <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> as the (nonstrict) <a class='existingWikiWord' href='/nlab/show/diff/localization'>localization</a> of <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> with respect to the class <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>.</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/dense+functor'>dense</a>.</p> </li> </ol> </div> <p>The equivalence of statements (1), (2), (4) and (5) are originally due to (<a href='#GabrielZisman67'>Gabriel-Zisman 67, Prop. 1.3, page 7</a>). The equivalence of (1) and (6) is due to (<a href='#Ulmer68'>Ulmer, Theorem 1.13</a>). The equivalence of (2) and (3) is (<a href='#Johnstone'>Johnstone, Lemma A1.1.1</a>).</p> <div class='proof'> <h6 id='proof'>Proof</h6> <p>The equivalence of (1) and (2) is <a href='adjoint+functor#FullyFaithfulAndInvertibleAdjoints'>this proposition</a>. The equivalence of (1) and (4) is <a href='idempotent+monad#EquivalentConditions'>this Prop.</a>. For (5) see <em><a class='existingWikiWord' href='/nlab/show/diff/reflective+localization'>reflective localization</a></em>. The equivalence of (1) and (6) can be seen by observing that <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>lan</mi> <mi>L</mi></msub><mi>L</mi><mo>≅</mo><mi>L</mi><msub><mi>lan</mi> <mi>L</mi></msub><mi>id</mi><mo>≅</mo><mi>L</mi><mi>R</mi></mrow><annotation encoding='application/x-tex'>lan_L L \cong L lan_L id \cong L R</annotation></semantics></math>, which is pointwise, since <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>lan</mi> <mi>L</mi></msub><mi>id</mi></mrow><annotation encoding='application/x-tex'>lan_L id</annotation></semantics></math> is absolute, and is isomorphic to the identity if and only if <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is fully faithful.</p> <p>To prove that (3) implies (2), the argument is to transfer the comonad structure on <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>R</mi></mrow><annotation encoding='application/x-tex'>L R</annotation></semantics></math> across the isomorphism to a comonad structure on <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>B</mi></msub></mrow><annotation encoding='application/x-tex'>1_B</annotation></semantics></math>, and observe that for any comonad structure on <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>B</mi></msub></mrow><annotation encoding='application/x-tex'>1_B</annotation></semantics></math> the counit is inverse to the comultiplication; thus the counit <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ε</mi></mrow><annotation encoding='application/x-tex'>\varepsilon</annotation></semantics></math> of the original comonad structure on <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>R</mi></mrow><annotation encoding='application/x-tex'>L R</annotation></semantics></math> must have been invertible. The same argument shows that for a comonad in any 2-category the counit <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ε</mi><mo>:</mo><mi>L</mi><mi>R</mi><mo>→</mo><msub><mn>1</mn> <mi>B</mi></msub></mrow><annotation encoding='application/x-tex'>\varepsilon : L R \to 1_B</annotation></semantics></math> is an isomorphism iff <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>R</mi></mrow><annotation encoding='application/x-tex'>L R</annotation></semantics></math> is isomorphic to <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>B</mi></msub></mrow><annotation encoding='application/x-tex'>1_B</annotation></semantics></math>.</p> </div> <p>This is a well-known set of equivalences concerning <a class='existingWikiWord' href='/nlab/show/diff/idempotent+monad'>idempotent monads</a>. The essential point is that a reflective subcategory <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>i \colon B \to A</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/monadic+functor'>monadic</a> (Prop. \ref{ReflectiveSubcategoryInclusionIsMonadic}), i.e., realizes <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> as the category of <a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+monad'>algebras for the monad</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mi>r</mi></mrow><annotation encoding='application/x-tex'>i r</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>r</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>r: A \to B</annotation></semantics></math> is the reflector.</p> <p>See also the related discussion at <a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(infinity,1)-category</a>.</p> <h2 id='special_cases'>Special cases</h2> <h3 id='ExactReflectiveSubcategories'>Exact reflective subcategories</h3> <p>If the reflector (which as a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> always preserves all <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a>s) in addition preserves <a class='existingWikiWord' href='/nlab/show/diff/finite+limit'>finite limits</a>, then the embedding is called <em>exact</em>. If the categories are <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a>es then such embeddings are called <a class='existingWikiWord' href='/nlab/show/diff/geometric+embedding'>geometric embedding</a>s.</p> <p>In particular, every <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>sheaf topos</a> is an exact reflective subcategory of a <a class='existingWikiWord' href='/nlab/show/diff/category+of+presheaves'>category of presheaves</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mover><mo>↪</mo><mover><mo>←</mo><mi>sheafify</mi></mover></mover><mi>PSh</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Sh(C) \stackrel{\overset{sheafify}{\leftarrow}}{\hookrightarrow} PSh(C) \,. </annotation></semantics></math></div> <p>The reflector in that case is the <a class='existingWikiWord' href='/nlab/show/diff/sheafification'>sheafification</a> functor.</p> <div class='num_theorem'> <h6 id='theorem'>Theorem</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a reflective subcategory of a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed category</a>, then it is an <a class='existingWikiWord' href='/nlab/show/diff/exponential+ideal'>exponential ideal</a> if and only if its <a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflector</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>D\to C</annotation></semantics></math> preserves <a class='existingWikiWord' href='/nlab/show/diff/finite+product'>finite products</a>.</p> <p>In particular, <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is then also cartesian closed.</p> </div> <p>This appears for instance as (<a href='#Johnstone'>Johnstone, A4.3.1</a>). See also at <em><a class='existingWikiWord' href='/nlab/show/diff/reflective+subuniverse'>reflective subuniverse</a></em>.</p> <p>So in particular if <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is an exact reflective subcategory of a cartesian closed category <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/exponential+ideal'>exponential ideal</a> of <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>.</p> <p>See <a class='existingWikiWord' href='/nlab/show/diff/Day%27s+reflection+theorem'>Day's reflection theorem</a> for a more general statement and proof.</p> <h3 id='complete_reflective_subcategories'>Complete reflective subcategories</h3> <p>When the unit of the reflector is a <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphism</a>, a reflective category is often thought of as a <a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full subcategory</a> of <em>complete</em> objects in some sense; the reflector takes each object in the ambient category to its completion. Such reflective subcategories are sometimes called <em>mono-reflective</em>. One similarly has <em>epi-reflective</em> (when the unit is an <a class='existingWikiWord' href='/nlab/show/diff/epimorphism'>epimorphism</a>) and <em>bi-reflective</em> (when the unit is both a monomorphism and an epimorphism).</p> <p>In the last case, note that if the unit is an <em>iso</em>morphism, then the inclusion functor is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalence of categories</a>, so nontrivial bireflective subcategories can occur only in non-<a class='existingWikiWord' href='/nlab/show/diff/balanced+category'>balanced categories</a>. Also note that ‘bireflective’ here does <em>not</em> mean reflective and <a class='existingWikiWord' href='/nlab/show/diff/coreflective+subcategory'>coreflective</a>. One sees this term often in discussions of <a class='existingWikiWord' href='/nlab/show/diff/concrete+category'>concrete categories</a> (such as <a class='existingWikiWord' href='/nlab/show/diff/topological+concrete+category'>topological categories</a>) where really something stronger holds: that the reflector lies over the <a class='existingWikiWord' href='/nlab/show/diff/identity+functor'>identity functor</a> on <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>. In this case, one can say that we have a subcategory that is <strong>reflective over <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math></strong>.</p> <h3 id='AccessibleReflectiveSubcategories'>Accessible reflective subcategories</h3> <div class='num_defn' id='AccessibleReflection'> <h6 id='definition_4'>Definition</h6> <p>A reflection</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mover><munder><mo>↪</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'> \mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} \mathcal{D} </annotation></semantics></math></div> <p>is called <strong>accessible</strong> if <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>\mathcal{D}</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/accessible+category'>accessible category</a> and the reflector <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>∘</mo><mi>L</mi><mo lspace='verythinmathspace'>:</mo><mi>𝒟</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>R\circ L \colon \mathcal{D} \to \mathcal{D}</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/accessible+functor'>accessible functor</a>.</p> </div> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>A reflective subcategory <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo>↪</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C} \hookrightarrow \mathcal{D}</annotation></semantics></math> of an <a class='existingWikiWord' href='/nlab/show/diff/accessible+category'>accessible category</a> is accessible, def. <a class='maruku-ref' href='#AccessibleReflection'>3</a>, precisely if <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/accessible+category'>accessible category</a>.</p> </div> <p>In this explicit form this appears as (<a href='#Lurie'>Lurie, prop. 5.5.1.2</a>). From (<a href='#AdamekRosicky'>Adamek-Rosický</a>) the “only if”-direction follows immediately from 2.53 there (saying that an accessibly embedded subcategory of an accessible category is accessible iff it is cone-reflective), while the “if”-direction follows immediately from 2.23 (saying any left or right adjoint between accessible categories is accessible).</p> <h2 id='properties'>Properties</h2> <h3 id='general'>General</h3> <p>A reflective subcategory is always closed under <a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a> which exist in the ambient category (because the full inclusion is monadic, by Prop. \ref{ReflectiveSubcategoryInclusionIsMonadic}, <a href='monadic+functor#MonadicFunctorsCreateLimits'>hence</a> creates limits, as noted above), and inherits <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a> from the larger category by application of the reflector <a href='#Riehl'>Riehl, Prop 4.5.15</a>. In particular, if the ambient category is complete and cocomplete then so is the reflective subcategory.</p> <p>A morphism in a reflective subcategory is monic iff it is monic in the ambient category. A reflective subcategory of a well-powered category is well-powered.</p> <h3 id='AsEilenbergMooreCategory'>As Eilenberg-Moore category of the idempotent monad</h3> <p>\begin{proposition} \label{ReflectiveSubcategoryInclusionIsMonadic} Every reflective subcategory inclusion is a <a class='existingWikiWord' href='/nlab/show/diff/monadic+functor'>monadic functor</a>, exhibiting the reflective subcategory as the <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Moore+category'>Eilenberg-Moore category</a> of <a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+monad'>modules</a> for its <a href='monad#RelationBetweenAdjunctionsAndMonads'>induced</a> <a class='existingWikiWord' href='/nlab/show/diff/idempotent+monad'>idempotent monad</a>. Conversely, the <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Moore+category'>Eilenberg-Moore category</a> of an <a class='existingWikiWord' href='/nlab/show/diff/idempotent+monad'>idempotent monad</a> is a reflective subcategory \end{proposition} A proof is spelled out for instance in <a href='#Borceux94b'>Borceux 1994, vol 2, cor. 4.2.4</a>. A <a class='existingWikiWord' href='/nlab/show/diff/proof'>formal proof</a> in <a class='existingWikiWord' href='/nlab/show/diff/Agda'>cubical Agda</a> is given in <a href='#1Lab'>1Lab</a>. See also Prop. <a class='maruku-ref' href='#CharacterizationByLocalization'>1</a> and see at <em><a href='idempotent+monad#AlgebrasForAnIdempotentMonad'>idempotent monad – Properties – Algebras for an idempotent monad and localization</a></em>.</p> <h3 id='reflective_subcategories_of_locally_presentable_categories'>Reflective subcategories of locally presentable categories</h3> <p>Both the weak and strong versions of <a class='existingWikiWord' href='/nlab/show/diff/Vop%C4%9Bnka%27s+principle'>Vopěnka's principle</a> are equivalent to fairly simple statements concerning reflective subcategories of locally presentable categories:</p> <div class='num_theorem'> <h6 id='theorem_2'>Theorem</h6> <p>The weak <a class='existingWikiWord' href='/nlab/show/diff/Vop%C4%9Bnka%27s+principle'>Vopěnka's principle</a> is equivalent to the statement:</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+category'>locally presentable category</a>, every <a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full subcategory</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>↪</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>D \hookrightarrow C</annotation></semantics></math> which is closed under <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>s is a reflective subcategory.</p> </div> <p>This is <a href='#AdamekRosicky'>AdamekRosicky, theorem 6.28</a></p> <div class='num_theorem'> <h6 id='theorem_3'>Theorem</h6> <p>The strong <a class='existingWikiWord' href='/nlab/show/diff/Vop%C4%9Bnka%27s+principle'>Vopěnka's principle</a> is equivalent to:</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+category'>locally presentable category</a>, every <a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full subcategory</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>↪</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>D \hookrightarrow C</annotation></semantics></math> which is closed under <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>s is a reflective subcategory; further on, <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is then also locally presentable.</p> </div> <p>(Remark after corollary 6.24 in <a href='#AdamekRosicky'>Adamek-Rosicky book</a>).</p> <h3 id='ReflectiveSubcategoriesOfCartesianClosedCategotries'>Reflective subcategories of cartesian closed categories</h3> <p>In showing that a given category is <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed</a>, the following theorem is often useful (cf. A4.3.1 in the <a class='existingWikiWord' href='/nlab/show/diff/Sketches+of+an+Elephant'>Elephant</a>):</p> <div class='num_theorem'> <h6 id='theorem_4'>Theorem</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is cartesian closed, and <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>⊆</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>D\subseteq C</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflective subcategory</a>, then the reflector <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo lspace='verythinmathspace'>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>L\colon C\to D</annotation></semantics></math> preserves finite <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>products</a> if and only if <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/exponential+ideal'>exponential ideal</a> (i.e. <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>Y\in D</annotation></semantics></math> implies <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Y</mi> <mi>X</mi></msup><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>Y^X\in D</annotation></semantics></math> for any <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>X\in C</annotation></semantics></math>). In particular, if <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> preserves finite products, then <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is cartesian closed.</p> </div> <h3 id='reflective_and_coreflective_subcategories'>Reflective and coreflective subcategories</h3> <div class='num_theorem'> <h6 id='theorem_5'>Theorem</h6> <p>A subcategory of a <a class='existingWikiWord' href='/nlab/show/diff/category+of+presheaves'>category of presheaves</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>A</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[A^{op}, Set]</annotation></semantics></math> which is both reflective and coreflective is itself a category of presheaves <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>B</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[B^{op}, Set]</annotation></semantics></math>, and the inclusion is induced by a functor <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A \to B</annotation></semantics></math>.</p> </div> <p>This is shown in (<a href='#BashirVelebil'>BashirVelebil</a>).</p> <h3 id='property_vs_structure'>Property vs structure</h3> <p>Whenever <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a full subcategory of <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>, we can say that objects of <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> are objects of <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> with some extra <a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>property</a>. But if <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is reflective in <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>, then we can turn this around and (by thinking of the left adjoint as a <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a>) think of objects of <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> as objects of <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> with (if we're lucky) some <a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>extra structure</a> or (in any case) some <a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>extra stuff</a>.</p> <p>This can always be made to work by brute force, but sometimes there is something insightful about it. For example, a metric space is a complete metric space equipped with a dense subset. Or, an <a class='existingWikiWord' href='/nlab/show/diff/integral+domain'>integral domain</a> is a <a class='existingWikiWord' href='/nlab/show/diff/field'>field</a> equipped with numerator and denominator functions.</p> <h2 id='Examples'>Examples</h2> <div class='num_example'> <h6 id='example'>Example</h6> <p>Complete <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric spaces</a> are mono-reflective in metric spaces; the reflector is called <em>completion</em>.</p> </div> <div class='num_example'> <h6 id='example_2'>Example</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/category+of+sheaves'>category of sheaves</a> on a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is a reflective subcategory of the <a class='existingWikiWord' href='/nlab/show/diff/category+of+presheaves'>category of presheaves</a> on <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>; the reflector is called <em><a class='existingWikiWord' href='/nlab/show/diff/sheafification'>sheafification</a></em>. In fact, categories of sheaves are precisely those accessible reflective subcategories, def. <a class='maruku-ref' href='#AccessibleReflection'>3</a>, of presheaf categories for which the reflector is left <a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>exact</a>. This makes the inclusion functor precisely a <a class='existingWikiWord' href='/nlab/show/diff/geometric+embedding'>geometric inclusion</a> of <a class='existingWikiWord' href='/nlab/show/diff/topos'>toposes</a>.</p> </div> <div class='num_example'> <h6 id='example_3'>Example</h6> <p>A category of <a class='existingWikiWord' href='/nlab/show/diff/concrete+sheaf'>concrete presheaves</a> inside a <a class='existingWikiWord' href='/nlab/show/diff/category+of+presheaves'>category of presheaves</a> on a <a class='existingWikiWord' href='/nlab/show/diff/concrete+site'>concrete site</a> is a reflective subcategory.</p> </div> <div class='num_example'> <h6 id='example_4'>Example</h6> <p>In a <a class='existingWikiWord' href='/nlab/show/diff/recollement'>recollement</a> situation, we have several reflectors and coreflectors. We have a reflective and coreflective subcategory <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mo>*</mo></msub><mo>:</mo><mi>A</mi><mo>′</mo><mo>↪</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>i_*: A' \hookrightarrow A</annotation></semantics></math> with reflector <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>i^*</annotation></semantics></math> and coreflector <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>i</mi> <mo>!</mo></msup></mrow><annotation encoding='application/x-tex'>i^!</annotation></semantics></math>. The functor <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>j</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>j^*</annotation></semantics></math> is both a reflector for the reflective subcategory <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>j</mi> <mo>*</mo></msub><mo>:</mo><mi>A</mi><mo>″</mo><mo>↪</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>j_*: A'' \hookrightarrow A</annotation></semantics></math>, and a coreflector for the coreflective subcategory <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>j</mi> <mo>!</mo></msub><mo>:</mo><mi>A</mi><mo>″</mo><mo>↪</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>j_!: A'' \hookrightarrow A</annotation></semantics></math>.</p> </div> <div class='num_example' id='TheReflectiveSubcategoriesOfSet'> <h6 id='example_5'>Example</h6> <p>Assuming classical logic, the category <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> has exactly three reflective (and <a class='existingWikiWord' href='/nlab/show/diff/replete+subcategory'>replete</a>) subcategories: the full subcategory containing all <a class='existingWikiWord' href='/nlab/show/diff/singleton'>singleton sets</a>; the full subcategory containing all <a class='existingWikiWord' href='/nlab/show/diff/subsingleton'>subsingletons</a>; and <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math> itself.</p> <p>In <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a>, there are potentially more reflective subcategories, for instance the subcategory of <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math>-sheaves for any <a class='existingWikiWord' href='/nlab/show/diff/Lawvere-Tierney+topology'>Lawvere-Tierney topology</a> on <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math>.</p> </div> <div class='num_example' id='AffineVarieties'> <h6 id='example_6'>Example</h6> <p>The category of <a class='existingWikiWord' href='/nlab/show/diff/affine+scheme'>affine schemes</a> is a reflective subcategory of the category of <a class='existingWikiWord' href='/nlab/show/diff/scheme'>schemes</a>, with the reflector given by <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>↦</mo><mi>Spec</mi><mi>Γ</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \mapsto Spec \Gamma(X,\mathcal{O}_X)</annotation></semantics></math>.</p> <p>The generalization of this example to <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> is discussed at <em><a class='existingWikiWord' href='/nlab/show/diff/function+algebras+on+infinity-stacks'>function algebras on infinity-stacks</a></em>. The analogue in <a class='existingWikiWord' href='/nlab/show/diff/noncommutative+algebraic+geometry'>noncommutative algebraic geometry</a> is in (<a href='#Rosenberg98'>Rosenberg 98, prop 4.4.3</a>).</p> </div> <div class='num_example' id='NonUnitalRings'> <h6 id='counterexample'>(Counter)Example</h6> <p>The non-full inclusion of unital <a class='existingWikiWord' href='/nlab/show/diff/ring'>rings</a> into non-unital rings has a left adjoint (with monic units), whose reflector formally adjoins an <a class='existingWikiWord' href='/nlab/show/diff/identity+element'>identity element</a>. However, we do not call it a reflective subcategory, because the “inclusion” is not full; see remark <a class='maruku-ref' href='#NonFullReflections'>1</a>.</p> </div> <div class='num_remark' id='RemarkOnNonUnitalRings'> <h6 id='remark_3'>Remark</h6> <p>Notice that for <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>∈</mo><mi>Ring</mi></mrow><annotation encoding='application/x-tex'>R \in Ring</annotation></semantics></math> a ring with unit, its reflection <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mi>R</mi></mrow><annotation encoding='application/x-tex'>L R</annotation></semantics></math> in the above example is not in general isomorphic to <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, but is much larger. But an object in a reflective subcategory is necessarily isomorphic to its image under the reflector only if the reflective subcategory is full. While the inclusion <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>Ring</mi></mstyle><mo>↪</mo><mstyle mathvariant='bold'><mi>Ring</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{Ring} \hookrightarrow \mathbf{Ring}</annotation></semantics></math>‘ does have a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> (as any <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> between varieties of algebras, by the <a class='existingWikiWord' href='/nlab/show/diff/adjoint+lifting+theorem'>adjoint lifting theorem</a>), this inclusion is not full (an arrow in <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>Ring</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{Ring}</annotation></semantics></math>’ need not preserve the identity).</p> </div> <div class='num_example' id='CategoryOfCategories'> <h6 id='example_7'>Example</h6> <p>The subcategory</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi><mo>↪</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'> Cat \hookrightarrow sSet</annotation></semantics></math></div> <p>of the <a class='existingWikiWord' href='/nlab/show/diff/Cat'>category of categories</a> into the <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>category of simplical sets</a> is a reflective subcategory <a href='#Riehl'>Riehl, example 4.5.14 (vi)</a>. The reflection is given by the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category functor</a>. This implies that <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a> is complete and cocomplete because it inherits all limits and colimits from <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>.</p> </div> <div class='num_example' id='ModelsOfALawvereTheory'> <h6 id='example_8'>Example</h6> <p>For any <a class='existingWikiWord' href='/nlab/show/diff/Lawvere+theory'>Lawvere theory</a> <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>, its category of models is the category</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Prod</mi><mo stretchy='false'>(</mo><mi>T</mi><mo>,</mo><mi>Set</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Prod(T, Set)</annotation></semantics></math></div> <p>of product preserving functors into <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math> and natural transformations between them. The inclusion</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Prod</mi><mo stretchy='false'>(</mo><mi>T</mi><mo>,</mo><mi>Set</mi><mo stretchy='false'>)</mo><mo>↪</mo><mo stretchy='false'>[</mo><mi>T</mi><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>Prod(T, Set) \hookrightarrow [T, Set]</annotation></semantics></math></div> <p>is a reflective subcategory <a href='#Buckley'>Buckley, theorem 5.2.1</a>. Therefore, because <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>T</mi><mo>,</mo><mi>Set</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[T,Set]</annotation></semantics></math> is complete and cocomplete (limits and colimits are computed pointwise), so is <math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Prod</mi><mo stretchy='false'>(</mo><mi>T</mi><mo>,</mo><mi>Set</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Prod(T, Set)</annotation></semantics></math>. This implies that many familar algebraic categories such as <a class='existingWikiWord' href='/nlab/show/diff/Grp'>Grp</a>, <a class='existingWikiWord' href='/nlab/show/diff/category+of+monoids'>Mon</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ring'>Ring</a>, etc. are complete and cocomplete as a special case.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization'>localization</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+category'>locally presentable category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+localization'>reflective localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Quillen+reflection'>Quillen reflection</a>, <a class='existingWikiWord' href='/nlab/show/diff/left+Bousfield+localization'>left Bousfield localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(infinity,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/coreflective+subcategory'>coreflective subcategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bireflective+subcategory'>bireflective subcategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+modality'>adjoint cylinder</a>, describing the situation when the reflector has a further left adjoint</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sheaf+toposes+are+equivalently+the+left+exact+reflective+subcategories+of+presheaf+toposes'>sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes</a></p> </li> </ul> <h2 id='references'>References</h2> <ul> <li id='GabrielZisman67'> <p><a class='existingWikiWord' href='/nlab/show/diff/Pierre+Gabriel'>Pierre Gabriel</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michel+Zisman'>Michel Zisman</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Calculus+of+fractions+and+homotopy+theory'>Calculus of fractions and homotopy theory</a></em>, Springer 1967 (<a href='https://people.math.rochester.edu/faculty/doug/otherpapers/GZ.pdf'>pdf</a>)</p> </li> <li id='Ulmer68'> <p><a class='existingWikiWord' href='/nlab/show/diff/Friedrich+Ulmer'>Friedrich Ulmer</a>, <em>Properties of Dense and Relative Adjoint Functors</em>, Journal of Algebra 1968</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, §IV.3 of: <em><a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]</p> </li> <li id='Borceux94a'> <p><a class='existingWikiWord' href='/nlab/show/diff/Francis+Borceux'>Francis Borceux</a>, p. 196 in <em>Basic Category Theory</em>, vol.1 of: <em><a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Categorical+Algebra'>Handbook of Categorical Algebra</a></em>, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]</p> </li> <li id='Borceux94b'> <p><a class='existingWikiWord' href='/nlab/show/diff/Francis+Borceux'>Francis Borceux</a>, Cor. 2.4.2 in: <em>Categories and Structures</em>, <em><a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Categorical+Algebra'>Handbook of Categorical Algebra</a></em>, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]</p> </li> <li id='AdamekRosicky'> <p><a class='existingWikiWord' href='/nlab/show/diff/Ji%C5%99%C3%AD+Ad%C3%A1mek'>Jiri Adamek</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ji%C5%99%C3%AD+Rosick%C3%BD'>Jiří Rosický</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Locally+Presentable+and+Accessible+Categories'>Locally presentable and accessible categories</a></em> London Mathematical Society Lecture Note Series 189</p> </li> <li> <p>Springer eom: <a href='http://eom.springer.de/r/r080550.htm'>reflective subcategory</a></p> </li> <li> <p><span><del class='diffmod'> cf.</del><ins class='diffmod'> C.</ins><del class='diffmod'> the</del><ins class='diffmod'> Cassidy,</ins><del class='diffmod'> notion</del><ins class='diffmod'> M.</ins><del class='diffmod'> of</del><ins class='diffmod'> Hébert,</ins></span><del class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_e3d86127617ec7452f161ed717b3a8bc2915d823_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Q</mi> <mo>∘</mo></msup></mrow><annotation encoding='application/x-tex'>Q^\circ</annotation></semantics></math></del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Max+Kelly'>G. M. Kelly</a></ins><span><del class='diffmod'> -category</del><ins class='diffmod'> ,</ins><del class='diffdel'> in</del><del class='diffdel'> the</del><del class='diffdel'> entry</del></span><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Q-category'>Q-category</a></del><ins class='diffmod'><em>Reflective subcategories, localizations and factorizationa systems</em></ins><ins class='diffins'>, J. Austral. Math. Soc. </ins><ins class='diffins'><strong>38</strong></ins><ins class='diffins'>:3 (1985) 287–329 </ins><ins class='diffins'><a href='https://doi.org/10.1017/S1446788700023624'>doi</a></ins></p> </li><ins class='diffins'> </ins><ins class='diffins'><li> <p>cf. the notion of <math class='maruku-mathml' display='inline' id='mathml_6d9ee3546c71ae0b81e8b4473f1d833632e07670_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Q</mi> <mo>∘</mo></msup></mrow><annotation encoding='application/x-tex'>Q^\circ</annotation></semantics></math>-category in the entry <a class='existingWikiWord' href='/nlab/show/diff/Q-category'>Q-category</a></p> </li></ins> <li id='Riehl'> <p><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Category+Theory+in+Context'>Category Theory in Context</a>, p. 142, Courier Dover Publications 2017 (<a href='http://www.math.jhu.edu/~eriehl/context.pdf'>pdf</a>)</em></p> </li> <li id='Buckley'> <p>Mitchell Buckley, <em>Lawvere Theories, 2008 <a href='http://web.science.mq.edu.au/~street/MitchB.pdf'>pdf</a></em></p> </li> </ul> <p>The relation of exponential ideals to <a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflective subcategories</a> is discussed in section A4.3.1 of</p> <ul> <li id='Johnstone'><a class='existingWikiWord' href='/nlab/show/diff/Peter+Johnstone'>Peter Johnstone</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Sketches+of+an+Elephant'>Sketches of an Elephant</a></em></li> </ul> <p>Reflective and coreflective subcategories of presheaf categories are discussed in</p> <ul> <li id='BashirVelebil'>R. Bashir, J. Velebil, <em>Simultaneously reflective and coreflective subcategories of presheaves</em>, Theory and Applications of Categories, Vol 10. No. 16. (2002) (<a href='http://www.emis.de/journals/TAC/volumes/10/16/10-16.pdf'>pdf</a>).</li> </ul> <p>Related discussion of <a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-categories</a> is in</p> <ul> <li id='Lurie'><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>Higher Topos Theory</a></em></li> </ul> <p>The example of affine schemes in <a class='existingWikiWord' href='/nlab/show/diff/noncommutative+algebraic+geometry'>noncommutative algebraic geometry</a> is in</p> <ul> <li id='Rosenberg98'><a class='existingWikiWord' href='/nlab/show/diff/Alexander+Rosenberg'>Alexander Rosenberg</a>, <em>Noncommutative schemes</em>, Comp. Math. 112, 93–125 (1998)</li> </ul> <p>Formalization in <a class='existingWikiWord' href='/nlab/show/diff/Agda'>cubical Agda</a>:</p> <ul> <li id='1Lab'><a class='existingWikiWord' href='/nlab/show/diff/1lab'>1lab</a>, <em><a href='https://1lab.dev/Cat.Functor.Adjoint.Reflective.html'>Reflective Subcategories</a></em></li> </ul> <p>See also:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Michael+Barr'>Michael Barr</a>, <a class='existingWikiWord' href='/nlab/show/diff/John+Kennison'>John Kennison</a>, <a class='existingWikiWord' href='/nlab/show/diff/Robert+Raphael'>Robert Raphael</a>, <em>On reflective and coreflective hulls</em>, <a class='existingWikiWord' href='/nlab/show/diff/Cahiers'>Cahiers Topologie Géométrie Différentielle Catégorique</a> <strong>56</strong> (2015) 162–208 [[pdf](https://www.math.mcgill.ca/barr/papers/biref.pdf), <a class='existingWikiWord' href='/nlab/files/BarrKennisonRaphael-Reflective.pdf' title='pdf'>pdf</a>]</li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on August 31, 2024 at 22:36:26. 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