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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#general_definition'>General definition</a></li> <li><a href='#in_abelian_categories'>In abelian categories</a></li> <li><a href='#in_chain_complexes'>In chain complexes</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#InjectiveModules'>Injective modules</a></li> <ul> <li><a href='#sketch_of_proof'>Sketch of proof</a></li> </ul> <li><a href='#injective_abelian_groups'>Injective abelian groups</a></li> <li><a href='#in_toposes'>In toposes</a></li> <li><a href='#in_topological_spaces'>In topological spaces</a></li> <li><a href='#in_boolean_algebras'>In Boolean algebras</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#preservation_of_injective_objects'>Preservation of injective objects</a></li> <li><a href='#ExistenceOfEnoughInjectives'>Existence of enough injectives</a></li> <li><a href='#InjectiveResolutions'>Injective resolutions</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>There is a very general notion of <em>injective objects</em> 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>, and then there is a sequence of more concrete notions as <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 equipped with more <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">structure and property</a>, in particular 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 <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> <a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">or a relative</a> thereof.</p> <p>The concept of <a class="existingWikiWord" href="/nlab/show/resolutions">resolutions</a> by injective objects – <a class="existingWikiWord" href="/nlab/show/injective+resolutions">injective resolutions</a> – is crucial notably in the discussion of <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> (in the context of <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a>: <a class="existingWikiWord" href="/nlab/show/derived+functors+in+homological+algebra">derived functors in homological algebra</a>).</p> <p>Being injective is a <em>property</em> of an object; the corresponding <em>structure</em> is called an <a class="existingWikiWord" href="/nlab/show/algebraic+injective">algebraic injective</a>.</p> <h3 id="general_definition">General definition</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>⊂</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J \subset Mor(C)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/class">class</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</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>.</p> <div class="num_example"> <h6 id="example">Example</h6> <p>Frequently <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is the class of all <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> or a related class.</p> <p>This is notably the case 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> is a <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> equipped with the injective <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is its class of <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a>.</p> </div> <div class="num_defn" id="InjectiveObjects"> <h6 id="definition_2">Definition</h6> <p>An object <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> 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> is <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-injective</strong> if all diagrams of the form</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><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>I</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow \\ Z } </annotation></semantics></math></div> <p>admit an <a class="existingWikiWord" href="/nlab/show/extension">extension</a></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><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>I</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mo>∃</mo></mpadded></msub></mtd></mtr> <mtr><mtd><mi>Z</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow & \nearrow_{\mathrlap{\exists}} \\ Z } \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is the class of all <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>, we speak merely of an <strong>injective object</strong>.</p> </div> <p> <div class='num_remark' id='InjectiveObjectsAsInjectiveMorphisms'> <h6>Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math>, then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-injective objects <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> according to def. <a class="maruku-ref" href="#InjectiveObjects"></a> are those for which the unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mover><mo>→</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover><mo>*</mo></mrow><annotation encoding="application/x-tex">I \stackrel{\exists!}{\to} \ast</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+morphism">injective morphism</a>.</p> </div> </p> <div class="num_defn" id="EnoughInjectives"> <h6 id="definition_3">Definition</h6> <p>One says that a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has <strong>enough injectives</strong> if every object admits a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> into an injective object.</p> </div> <p>(The <a class="existingWikiWord" href="/nlab/show/formal+duality">dual</a> notion is that of a <em><a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></em>.)</p> <p>Assuming the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>, we have the following easy result:</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Any <a class="existingWikiWord" href="/nlab/show/small+set">small</a> <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of injective objects is injective.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math> then these extensions are equivalently <a class="existingWikiWord" href="/nlab/show/lifts">lifts</a></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><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>I</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mrow></mrow> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msup></mrow></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mo>∃</mo></mpadded></msub></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mi>Z</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& I \\ \mathllap{{}^{j \in J}} \big\downarrow & \nearrow_{\mathrlap{\exists}} & \big\downarrow \\ Z &\longrightarrow& * } </annotation></semantics></math></div> <p>and hence the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-injective objects are precisely those that have the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> then <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-injective precisely if the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a></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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><msup><mi>C</mi> <mi>op</mi></msup><mo>⟶</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> Hom_C(-,I) \,\colon\, C^{op} \longrightarrow Set </annotation></semantics></math></div> <p>takes morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>s in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> </div> <h3 id="in_abelian_categories">In abelian categories</h3> <p>The term <em>injective object</em> is used most frequently in the context that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <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> an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> of monomorphisms is the same as the class of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">0 \to A \stackrel{f}{\to} B</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By definition of <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> every monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow B</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, hence a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the form</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><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\to& 0 \\ \downarrow && \downarrow \\ B &\to& C } </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> the (algebraic) <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a>. By the <a href="http://ncatlab.org/nlab/show/pullback#Pasting">pasting law</a> for pullbacks we find that also the left square in</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><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& A &\to& 0 \\ \downarrow && \downarrow && \downarrow \\ 0 &\to& B &\to& C } </annotation></semantics></math></div> <p>is a pullback, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">0 \to A \to B</annotation></semantics></math> is exact.</p> </div> <div class="num_cor" id="EquivalentCharacterizationOfInjectivesInAbelianCategories"> <h6 id="corollary">Corollary</h6> <p>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>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> of an abelian category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is then <strong>injective</strong> if it satisfies the following equivalent conditions:</p> <ul> <li> <p>the <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><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi class="mathscript">𝒜𝒷</mi></mrow><annotation encoding="application/x-tex">Hom_C(-, I) : C^{op} \to \mathscr{Ab}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/exact+functor">exact</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">𝒜𝒷</mi></mrow><annotation encoding="application/x-tex">\mathscr{Ab}</annotation></semantics></math> is the category of abelian groups;</p> </li> <li> <p>for all <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">0 \to X \to Y</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact</a> and for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">k : X \to I</annotation></semantics></math>, there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">h : Y \to I</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex"> h\circ f = k</annotation></semantics></math>.</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>k</mi></mpadded></msup></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo>∃</mo><mi>h</mi></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>I</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& X &\stackrel{f}{\to}& Y \\ && \downarrow^{\mathrlap{k}} & \swarrow_{\mathrlap{\exists h}} \\ && I } \,. </annotation></semantics></math></div></div> <p>By the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of <a href="projective+module#NProjectiveIFFHomNExact">this prop.</a>.</p> <h3 id="in_chain_complexes">In chain complexes</h3> <p>See <a class="existingWikiWord" href="/nlab/show/homotopically+injective+object">homotopically injective object</a> for a relevant generalization to categories of chain complexes, and its relationship to ordinary injectivity.</p> <h2 id="examples">Examples</h2> <h3 id="InjectiveModules">Injective modules</h3> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">C = R Mod</annotation></semantics></math> the category of <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/modules">modules</a>. We discuss <a class="existingWikiWord" href="/nlab/show/injective+modules">injective modules</a> over <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> (see there for more).</p> <p>The following criterion says that for identifying <a class="existingWikiWord" href="/nlab/show/injective+modules">injective modules</a> it is sufficient to test the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> which characterizes injective objects by def. <a class="maruku-ref" href="#InjectiveObjects"></a>, only on those <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> which include an <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> into the base ring <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>.</p> <div class="num_theorem" id="BaerTheorem"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a>)</strong></p> <p>If the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> holds, then a <a class="existingWikiWord" href="/nlab/show/module">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>∈</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">Q \in R Mod</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/injective+module">injective module</a> precisely if for <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> any left <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/ideal">ideal</a> regarded as an <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>-module, any <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">g : I \to Q</annotation></semantics></math> 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> can be extended to all of <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> along the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>↪</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">I \hookrightarrow R</annotation></semantics></math>.</p> </div> <p>This is due to (<a href="#Baer">Baer</a>).</p> <div class="proof"> <h6 id="sketch_of_proof">Sketch of proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>M</mi><mo>↪</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">i \colon M \hookrightarrow N</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>M</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">f \colon M \to Q</annotation></semantics></math> be a map. We must extend <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> to a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>N</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">h \colon N \to Q</annotation></semantics></math>. Consider the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> whose elements are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>′</mo><mo>,</mo><mi>f</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(M', f')</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">M'</annotation></semantics></math> is an intermediate <a class="existingWikiWord" href="/nlab/show/submodule">submodule</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>M</mi><mo>′</mo><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">f' \colon M' \to Q</annotation></semantics></math> is an extension 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>, ordered by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>′</mo><mo>,</mo><mi>f</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≤</mo><mo stretchy="false">(</mo><mi>M</mi><mo>″</mo><mo>,</mo><mi>f</mi><mo>″</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(M', f') \leq (M'', f'')</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">M''</annotation></semantics></math> contains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">M'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">f''</annotation></semantics></math> extends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math>. By an application of <a class="existingWikiWord" href="/nlab/show/Zorn%27s+lemma">Zorn's lemma</a>, this poset has a <a class="existingWikiWord" href="/nlab/show/maximal+element">maximal element</a>, say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>′</mo><mo>,</mo><mi>f</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(M', f')</annotation></semantics></math>. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">M'</annotation></semantics></math> is not all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">x \in N</annotation></semantics></math> be an element not in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">M'</annotation></semantics></math>; we show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> extends to a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>″</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>x</mi><mo stretchy="false">⟩</mo><mo>+</mo><mi>M</mi><mo>′</mo><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">M'' = \langle x \rangle + M' \to Q</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/contradiction">contradiction</a>.</p> <p>The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>r</mi><mo>∈</mo><mi>R</mi><mo>:</mo><mi>r</mi><mi>x</mi><mo>∈</mo><mi>M</mi><mo>′</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{r \in R: r x \in M'\}</annotation></semantics></math> is an ideal <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> of <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>, and we have a module <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">g \colon I \to Q</annotation></semantics></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>r</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(r) = f'(r x)</annotation></semantics></math>. By <a class="existingWikiWord" href="/nlab/show/hypothesis">hypothesis</a>, we may extend <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> to a module map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="verythinmathspace">:</mo><mi>R</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">k \colon R \to Q</annotation></semantics></math>. Writing a general element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">M''</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mi>x</mi><mo>+</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">r x + y</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>M</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">y \in M'</annotation></semantics></math>, it may be shown that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>″</mo><mo stretchy="false">(</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f''(r x + y) = k(r) + f'(y)</annotation></semantics></math></div> <p>is well-defined and extends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math>, as desired.</p> </div> <div class="num_corollary" id="DirectSumInjectives"> <h6 id="corollary_2">Corollary</h6> <p>(Assume that the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> holds.) Let <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> be a <a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Q</mi> <mi>j</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{Q_j\}_{j \in J}</annotation></semantics></math> be a collection of <a class="existingWikiWord" href="/nlab/show/injective+modules">injective modules</a> over <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>. Then the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub><msub><mi>Q</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">Q = \bigoplus_{j \in J} Q_j</annotation></semantics></math> is also injective.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By Baer’s criterion, theorem <a class="maruku-ref" href="#BaerTheorem"></a>, it suffices to show that for any <a class="existingWikiWord" href="/nlab/show/ideal">ideal</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> of <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 module <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">f \colon I \to Q</annotation></semantics></math> extends to a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">R \to Q</annotation></semantics></math>. Since <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> is Noetherian, <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 <a class="existingWikiWord" href="/nlab/show/finitely+generated+module">finitely generated</a> as an <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>-module, say by elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1, \ldots, x_n</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>j</mi></msub><mo lspace="verythinmathspace">:</mo><mi>Q</mi><mo>→</mo><msub><mi>Q</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">p_j \colon Q \to Q_j</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/projection">projection</a>, and put <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>j</mi></msub><mo>=</mo><msub><mi>p</mi> <mi>j</mi></msub><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">f_j = p_j \circ f</annotation></semantics></math>. Then for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_j(x_i)</annotation></semantics></math> is nonzero for only finitely many summands. Taking all of these summands together over all <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>, we see that <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> factors through</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>j</mi><mo>∈</mo><mi>J</mi><mo>′</mo></mrow></munder><msub><mi>Q</mi> <mi>j</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi><mo>′</mo></mrow></munder><msub><mi>Q</mi> <mi>j</mi></msub><mo>↪</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q</annotation></semantics></math></div> <p>for some finite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>′</mo><mo>⊂</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">J' \subset J</annotation></semantics></math>. But a <a class="existingWikiWord" href="/nlab/show/product">product</a> of injectives is injective, hence <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> extends to a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>→</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi><mo>′</mo></mrow></msub><msub><mi>Q</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">R \to \prod_{j \in J'} Q_j</annotation></semantics></math>, which completes the proof.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Conversely, <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> is a <a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a> if <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a> of injective <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/modules">modules</a> are injective.</p> </div> <p>This is due to Bass and Papp. See (<a href="#Lam">Lam, Theorem 3.46</a>).</p> <h3 id="injective_abelian_groups">Injective abelian groups</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>ℤ</mi><mi>Mod</mi><mo>≃</mo></mrow><annotation encoding="application/x-tex">C = \mathbb{Z} Mod \simeq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> be the <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>.</p> <div class="num_prop" id="InjectiveAbelianGroupIsDivisibleGroup"> <h6 id="proposition_5">Proposition</h6> <p>If the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> holds, then an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an injective object in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> precisely if it is a <a class="existingWikiWord" href="/nlab/show/divisible+group">divisible group</a>, in that for all <a class="existingWikiWord" href="/nlab/show/integers">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mi>G</mi><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">n G = G</annotation></semantics></math>.</p> </div> <p>This follows for instance using <a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a>, prop. <a class="maruku-ref" href="#BaerTheorem"></a>.</p> <p>An explicit proof is spelled out at <a href="https://planetmath.org/abeliangroupisdivisibleifandonlyifitisaninjectiveobject">Planet math – abelian group is divisible if and only if it is an injective object</a></p> <div class="num_example"> <h6 id="example_2">Example</h6> <p>By prop. <a class="maruku-ref" href="#InjectiveAbelianGroupIsDivisibleGroup"></a> the following <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> are injective in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>.</p> <p>The group of <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> is injective in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>, as is the additive group of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> <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">\mathbb{R}</annotation></semantics></math> and generally that underlying any <a class="existingWikiWord" href="/nlab/show/field">field</a> of characteristic zero. The additive group underlying any <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> is injective. The <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of any injective group by any other group is injective.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p><strong>Not</strong> injective in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> is for instance the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/n\mathbb{Z}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \gt 1</annotation></semantics></math>.</p> </div> <h3 id="in_toposes">In toposes</h3> <p>In any <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, the <a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> is an injective object, as is any <a class="existingWikiWord" href="/nlab/show/powering">power</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Mac+Lane-Moerdijk">Mac Lane-Moerdijk</a>, IV.10).</p> <p>Also one can define various notions of <em>internally</em> injective objects. These turn out to be equivalent:</p> <div class="num_prop" id="EquivalenceOfInternalNotionsOfInjectivity"> <h6 id="proposition_6">Proposition</h6> <p>In any <a class="existingWikiWord" href="/nlab/show/elementary+topos">elementary topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a>, the following statements about an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∈</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">I \in \mathcal{E}</annotation></semantics></math> are equivalent.</p> <ol> <li>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>I</mi><mo stretchy="false">]</mo><mo>:</mo><msup><mi>ℰ</mi> <mi>op</mi></msup><mo>→</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">[-, I] : \mathcal{E}^op \to \mathcal{E}</annotation></semantics></math> maps monomorphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> to epimorphisms.</li> <li>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>I</mi><mo stretchy="false">]</mo><mo>:</mo><msup><mi>ℰ</mi> <mi>op</mi></msup><mo>→</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">[-, I] : \mathcal{E}^op \to \mathcal{E}</annotation></semantics></math> maps monomorphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> to morphisms for which any global element of the target locally (after <a class="existingWikiWord" href="/nlab/show/change+of+base">change of base</a> along an epimorphism) possesses a preimage.</li> <li>For any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>A</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p : A \to 1</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>, the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup><mi>I</mi></mrow><annotation encoding="application/x-tex">p^*I</annotation></semantics></math> has property 1. as an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}/A</annotation></semantics></math>.</li> <li>For any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>A</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p : A \to 1</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>, the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup><mi>I</mi></mrow><annotation encoding="application/x-tex">p^*I</annotation></semantics></math> has property 2. as an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}/A</annotation></semantics></math>.</li> <li>The interpretation of the statement “<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 an injective object” using the <a class="existingWikiWord" href="/nlab/show/stack+semantics">stack semantics</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> holds.</li> </ol> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>The implications “1. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> 2.”, “3 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> 4.”, “3. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> 1.”, and “4. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> 2.” are trivial.</p> <p>The equivalence “3. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Leftrightarrow</annotation></semantics></math> 5.” follows directly from the interpretation rules of the stack semantics.</p> <p>The implication “2. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> 4.” employs the <a class="existingWikiWord" href="/nlab/show/base+change#GeometricMorphism">extra left-adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub><mo>:</mo><mi>ℰ</mi><mo stretchy="false">/</mo><mi>A</mi><mo>→</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">p_! : \mathcal{E}/A \to \mathcal{E}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup><mo>:</mo><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">p^* : \mathcal{E} \to \mathcal{E}/A</annotation></semantics></math>, as in the usual proof that injective sheaves remain injective when restricted to smaller open subsets: We have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>*</mo></msub><mo>∘</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>p</mi> <mo>*</mo></msup><mi>I</mi><msub><mo stretchy="false">]</mo> <mrow><mi>ℰ</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub><mo>≅</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>I</mi><msub><mo stretchy="false">]</mo> <mi>ℰ</mi></msub><mo>∘</mo><msub><mi>p</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">p_* \circ [-, p^*I]_{\mathcal{E}/A} \cong [-, I]_{\mathcal{E}} \circ p_!</annotation></semantics></math>, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">p_!</annotation></semantics></math> preserves monomorphisms, and one can check that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">p_*</annotation></semantics></math> reflects the property that global elements locally possess preimages. Details are in (<a href="#Harting">Harting, Theorem 1.1</a>).</p> <p>The implication “4. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> 3.” follows by performing an extra change of base, since any non-global element becomes a global element after a suitable change of base.</p> </div> <p>Somewhat surprisingly, and in stark contrast with the situation for <a class="existingWikiWord" href="/nlab/show/internally+projective+objects">internally projective objects</a>, internal injectivity coincides with external injecticity.</p> <div class="num_prop" id="EquivalenceOfInternalAndExternalInjectivity"> <h6 id="proposition_7">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> be the topos of <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> over a <a class="existingWikiWord" href="/nlab/show/locale">locale</a>. Then an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∈</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">I \in \mathcal{E}</annotation></semantics></math> is internally injective (in any of the senses given by Proposition <a class="maruku-ref" href="#EquivalenceOfInternalNotionsOfInjectivity"></a>) if and only if <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 injective as in Definition <a class="maruku-ref" href="#InjectiveObjects"></a>.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Let <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> be an externally injective object. Then <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> satisfies condition 2. of Proposition <a class="maruku-ref" href="#EquivalenceOfInternalNotionsOfInjectivity"></a>, even without having to pass to a cover.</p> <p>Conversely, let <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> be an internally injective object. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">m : X \to Y</annotation></semantics></math> be a monomorphism and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">k : X \to I</annotation></semantics></math> be an arbitrary morphism. We want to show that there exists an extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">Y \to I</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math>. To this end, consider the sheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>≔</mo><mo stretchy="false">{</mo><mi>k</mi><mo>′</mo><mo>:</mo><mi>ℋ</mi><mi>om</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>k</mi><mo>′</mo><mo>∘</mo><mi>m</mi><mo>=</mo><mi>k</mi><mo stretchy="false">}</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> F \coloneqq \{ k' : \mathcal{H}om(Y,I) | k' \circ m = k \}. </annotation></semantics></math></div> <p>One can check that <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 <a class="existingWikiWord" href="/nlab/show/flabby+sheaf">flabby</a> (this is particularly easy using the <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a>, details will be added later) and therefore has a global section.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>The analogs of Proposition <a class="maruku-ref" href="#EquivalenceOfInternalNotionsOfInjectivity"></a> and Proposition <a class="maruku-ref" href="#EquivalenceOfInternalAndExternalInjectivity"></a> for abelian group objects instead of unstructured objects hold as well, with mostly the same proofs. Condition 1. then refers to the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>:</mo><mi>Ab</mi><mo stretchy="false">(</mo><mi>ℰ</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mi>Ab</mi><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[-, X] : Ab(\mathcal{E})^op \to Ab(\mathcal{E})</annotation></semantics></math>.</p> </div> <h3 id="in_topological_spaces">In topological spaces</h3> <p>In the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of all topological spaces, the injective objects are precisely the <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a> <a class="existingWikiWord" href="/nlab/show/indiscrete+spaces">indiscrete spaces</a>.</p> <p>In the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math> spaces (see <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a>), the injective objects are the terminal spaces.</p> <p>In the above two cases, this refers to injectivity with respect to monomorphisms.</p> <p>In the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math> spaces, the injective objects with respect to homeomorphic embeddings are precisely those given by <a class="existingWikiWord" href="/nlab/show/Scott+topologies">Scott topologies</a> on <a class="existingWikiWord" href="/nlab/show/continuous+lattices">continuous lattices</a>; as <a class="existingWikiWord" href="/nlab/show/locales">locales</a> these are <a class="existingWikiWord" href="/nlab/show/local+compactum">locally compact</a> and <a class="existingWikiWord" href="/nlab/show/spatial+locale">spatial</a>. (Such spaces are usually called, perhaps confusingly, <em>injective spaces</em>.)</p> <p>In the category of all spaces, the injectives with respect to homeomorphic embeddings (i.e., <a class="existingWikiWord" href="/nlab/show/regular+monomorphisms">regular monomorphisms</a>) are the spaces whose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math> reflections are continuous lattices under the Scott topology.</p> <h3 id="in_boolean_algebras">In Boolean algebras</h3> <p>Injective objects in the category of <a class="existingWikiWord" href="/nlab/show/Boolean+algebras">Boolean algebras</a> are precisely <a class="existingWikiWord" href="/nlab/show/complete+Boolean+algebras">complete Boolean algebras</a>. This is the dual form of a theorem of Gleason, saying that the <a class="existingWikiWord" href="/nlab/show/projective+objects">projective objects</a> in the category of <a class="existingWikiWord" href="/nlab/show/Stone+spaces">Stone spaces</a> are the <a class="existingWikiWord" href="/nlab/show/extremally+disconnected+topological+space">extremally disconnected</a> ones (the closure of every open set is again open).</p> <h2 id="properties">Properties</h2> <h3 id="preservation_of_injective_objects">Preservation of injective objects</h3> <div class="num_lemma" id="RightAdjointsOfExactFunctorsPreserveInjectives"> <h6 id="lemma">Lemma</h6> <p>Given a pair of <a class="existingWikiWord" href="/nlab/show/additive+functor">additive</a> <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a></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>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℬ</mi><mover><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mi>L</mi></mover></mover><mi>𝒜</mi></mrow><annotation encoding="application/x-tex"> (L \dashv R) \;\colon\; \mathcal{B} \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}} \mathcal{A} </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> such that the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a> (thus automatically exact), then the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> preserves injective objects.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>Observe that an object is injective precisely if the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> into it sends <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> to <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a>, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves monomorphisms by assumption of (left-)exactness. With this the statement follows via adjunction isomorphism</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>𝒜</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>ℬ</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{A}}(-,R(I))\simeq Hom_{\mathcal{B}}(L(-),I) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>Additivity of the left adjoint follows from the remaining assumptions, since <a class="existingWikiWord" href="/nlab/show/additive+functor#SufficientConditions">exact functors preserve biproducts</a>.</p> </div> <p>The preceding lemma has the following variant:</p> <div class="num_prop" id="Adjuncts_Injectives"> <h6 id="proposition_8">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, <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">\mathcal{D}</annotation></semantics></math> be categories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>⊣</mo><mi>R</mi><mo>:</mo><mi>𝒟</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">L\dashv R:\mathcal{D}\to\mathcal{C}</annotation></semantics></math> be an adjunction. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> maps monos to monos, 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> maps injectives to injectives.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">I\in\mathcal{D}</annotation></semantics></math> be injective. Consider the following diagram 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">\mathcal{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><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>↣</mo><mi>m</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><mi>f</mi><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A & \overset{m}{\rightarrowtail} & B \\ f\downarrow & & \\ R(I) & & } </annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>Hom</mi> <mi>𝒟</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\theta: Hom_\mathcal{C}(X,R(Y))\overset{\simeq}{\rightarrow}Hom_\mathcal{D}(L(X),Y)</annotation></semantics></math> the natural bijection given by the adjunction. Consider now the following diagram in <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">\mathcal{D}</annotation></semantics></math> where the assumptions ensure that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(m)</annotation></semantics></math> is mono:</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><mi>L</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↣</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>L</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>θ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>I</mi></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ L(A) & \overset{L(m)}{\rightarrowtail} & L(B) \\ \theta(f)\downarrow & & \\ I & & } </annotation></semantics></math></div> <p>Since <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 injective, there there exists a filler <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi>L</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\theta(g):L(B)\to I</annotation></semantics></math> which by the adjunction must come from a uniquely determined <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g:B\to R(I)</annotation></semantics></math>. But the naturality of the bijection with respect to composition says that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>↣</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>θ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover><mi>I</mi></mrow><mrow><mi>A</mi><mover><mo>↣</mo><mi>m</mi></mover><mi>B</mi><mover><mo>→</mo><mi>g</mi></mover><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \frac{L(A)\overset{L(m)}{\rightarrowtail} L(B)\overset{\theta(g)}{\rightarrow}I}{A\overset{m}{\rightarrowtail} B\overset{g}{\rightarrow}R(I)} </annotation></semantics></math></div> <p>correspond to each other under the bijection whence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>m</mi><mo stretchy="false">)</mo><mo>=</mo><mi>θ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>L</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\theta(g\circ m)=\theta(g)\circ L(m)</annotation></semantics></math> but from the commutativity of the second diagram we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>L</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>=</mo><mi>θ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>θ</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\theta(g)\circ L(m)=\theta(f)=\theta(g\circ m)</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is a bijection it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>g</mi><mo>∘</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">f=g\circ m</annotation></semantics></math> which proves 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>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(I)</annotation></semantics></math> is injective.</p> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>The proof transposes the proof of the dual statement 10.2. in (<a href="#HiltonStammbach71">Hilton-Stammbach 1971</a>, p. 82): In situation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>⊣</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">L\dashv R</annotation></semantics></math>, if <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> maps epis to epis then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> maps projectives to projectives.</p> </div> <div class="num_remark" id="Exponential_injectives"> <h6 id="remark_6">Remark</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X\in\mathcal{C}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mo>−</mo></msub><mo>×</mo><mo>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">{}_-\times:\mathcal{C}\to\mathcal{C}</annotation></semantics></math> exists and has a right adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mo>−</mo></msub><mo>×</mo><mi>X</mi><mo>⊣</mo><msub><mo stretchy="false">(</mo> <mo>−</mo></msub><msup><mo stretchy="false">)</mo> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">{}_-\times X\dashv (_-)^X</annotation></semantics></math>. Since it is easy to check that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mo>−</mo></msub><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">{}_-\times X</annotation></semantics></math> preserves monos it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo stretchy="false">(</mo> <mo>−</mo></msub><msup><mo stretchy="false">)</mo> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">(_-)^X</annotation></semantics></math> preserves injectives.</p> <p>In particular, for toposes this implies that all power objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^X</annotation></semantics></math> are injective since the injectivity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> follows more or less straightforwardly from its classifying properties.</p> </div> <h3 id="ExistenceOfEnoughInjectives">Existence of enough injectives</h3> <p>We discuss a list of classes of categories that have <em>enough injectives</em> according to def. <a class="maruku-ref" href="#EnoughInjectives"></a>.</p> <div class="num_prop"> <h6 id="proposition_9">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/topos">topos</a> has enough injectives.</p> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/power+object">power object</a> can be shown to be injective (cf. the above <a href="#Exponentials_injectives">remark</a>), and every object embeds into its power object by the “singletons” map.</p> </div> <div class="num_prop" id="AbHasEnoughInjectives"> <h6 id="proposition_10">Proposition</h6> <p>Assuming some form of the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>, the category of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> has enough injectives.</p> </div> <p>Full AC is much more than required, however; <a class="existingWikiWord" href="/nlab/show/small+violations+of+choice">small violations of choice</a> suffices.</p> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>By prop. <a class="maruku-ref" href="#InjectiveAbelianGroupIsDivisibleGroup"></a> an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> is an injective <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-module precisely if it is a <a class="existingWikiWord" href="/nlab/show/divisible+group">divisible group</a>. So we need to show that every <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> is a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of a <a class="existingWikiWord" href="/nlab/show/divisible+group">divisible group</a>.</p> <p>To start with, notice that the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> is divisible and hence the canonical embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>↪</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \hookrightarrow \mathbb{Q}</annotation></semantics></math> shows that the additive group of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> embeds into an injective <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-module.</p> <p>Now by the discussion at <em><a class="existingWikiWord" href="/nlab/show/projective+module">projective module</a></em> every <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> receives an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mo>⊕</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub><mi>ℤ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(\oplus_{s \in S} \mathbb{Z}) \to A</annotation></semantics></math> from a <a class="existingWikiWord" href="/nlab/show/free+group">free</a> abelian group, hence is the <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a> of a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>. Accordingly it embeds into a quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde A</annotation></semantics></math> of a direct sum of copies of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</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><mtr><mtd><mi>ker</mi></mtd> <mtd><mover><mo>→</mo><mo>=</mo></mover></mtd> <mtd><mi>ker</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msub><mo>⊕</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mo>⊕</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub><mi>ℚ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mover><mi>A</mi><mo stretchy="false">˜</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ ker &\stackrel{=}{\to}& ker \\ \downarrow && \downarrow \\ (\oplus_{s \in S} \mathbb{Z}) &\hookrightarrow& (\oplus_{s \in S} \mathbb{Q}) \\ \downarrow && \downarrow \\ A &\hookrightarrow& \tilde A } </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde A</annotation></semantics></math> is divisible because the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of divisible groups is again divisible, and also the <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a> of a divisible groups is again divisble. So this exhibits an embedding of any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> into a divisible abelian group, hence into an injective <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-module.</p> </div> <div class="num_lemma" id="TransferOfEnoughInjectivesAlongAdjunctions"> <h6 id="lemma_2">Lemma</h6> <p>Given a pair of <a class="existingWikiWord" href="/nlab/show/additive+functor">additive</a> <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a></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>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℬ</mi><mover><munder><mo>⟶</mo><mi>R</mi></munder><mover><mo>⟵</mo><mi>L</mi></mover></mover><mi>𝒜</mi></mrow><annotation encoding="application/x-tex"> (L \dashv R) \;\colon\; \mathcal{B} \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}} \mathcal{A} </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> such that the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a>,</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful functor</a>.</p> </li> </ol> <p>Then if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math> has enough injectives, also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> has enough injectives.</p> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{A}</annotation></semantics></math>. By the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math> has enough injectives, there is an injective object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∈</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">I \in \mathcal{B}</annotation></semantics></math> and a monomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i \colon L(A) \hookrightarrow I</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of this is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>i</mi><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⟶</mo><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde i \colon A \longrightarrow R(I) </annotation></semantics></math></div> <p>and so it is sufficient to show that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(I)</annotation></semantics></math> is injective in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>i</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde i</annotation></semantics></math> is a monomorphism.</p> </li> </ol> <p>The first point is the statement of lemma <a class="maruku-ref" href="#RightAdjointsOfExactFunctorsPreserveInjectives"></a>.</p> <p>For the second point, consider the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>i</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde i</annotation></semantics></math> as part of the <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>⟶</mo><mi>A</mi><mover><mo>⟶</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover></mover><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ker(\tilde i)\longrightarrow A \stackrel{\tilde i}{\longrightarrow} R(I) \,. </annotation></semantics></math></div> <p>By the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a>, the image of this sequence under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is still exact</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>ker</mi><mo stretchy="false">(</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><mi>L</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L(ker(\tilde i)) \longrightarrow L(A) \stackrel{L(\tilde i)}{\longrightarrow} L(R(I)) \,. </annotation></semantics></math></div> <p>Now observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\tilde i)</annotation></semantics></math> is a monomorphism: this is because its composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>ϵ</mi></mover><mi>I</mi></mrow><annotation encoding="application/x-tex">L(A) \stackrel{L(\tilde i)}{\longrightarrow} L(R(I)) \stackrel{\epsilon}{\longrightarrow} I</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> is (by the formula for <a class="existingWikiWord" href="/nlab/show/adjuncts">adjuncts</a>) the original morphism <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>, which by construction is a monomorphism. Therefore the exactness of the above sequence means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>ker</mi><mo stretchy="false">(</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>L</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(ker(\tilde i)) \to L(A)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a>; and by the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful functor</a> this means that already <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">ker(\tilde i) \to A</annotation></semantics></math> is zero, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ker</mi><mo stretchy="false">(</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">ker(\tilde i) = 0</annotation></semantics></math>, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>i</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde i</annotation></semantics></math> is a monomorphism.</p> </div> <div class="num_prop" id="RModHasEnoughInjectives"> <h6 id="proposition_11">Proposition</h6> <p>As soon as the category <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> has enough injectives, so does the <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> <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/Mod">Mod</a> of <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over some <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <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>.</p> <p>In particular if the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> holds, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math> has enough injectives.</p> </div> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>Observe that the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>R</mi><mi>Mod</mi><mo>→</mo><mi>AbGp</mi></mrow><annotation encoding="application/x-tex">U\colon R Mod \to AbGp</annotation></semantics></math> has both a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">R_!</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> to <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>) and a right adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">R_*</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/coextension+of+scalars">coextension of scalars</a>). Since it has a left adjoint, it is <a class="existingWikiWord" href="/nlab/show/exact+functor">exact</a>. Thus the statement follows via lemma <a class="maruku-ref" href="#TransferOfEnoughInjectivesAlongAdjunctions"></a> from prop. <a class="maruku-ref" href="#AbHasEnoughInjectives"></a>.</p> </div> <div class="num_prop"> <h6 id="proposition_12">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">R = k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a>, hence <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/Mod">Mod</a> = <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>, every object is both injective as well as <a class="existingWikiWord" href="/nlab/show/projective+object">projective</a>.</p> </div> <div class="num_prop" id="AbelianSheavesHaveEnoughProjectives"> <h6 id="proposition_13">Proposition</h6> <p>The category of <a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ab(Sh(C))</annotation></semantics></math> on any <a class="existingWikiWord" href="/nlab/show/small+site">small site</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>, hence the category of abelian groups in the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> over <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>, has enough injectives.</p> </div> <p>A proof of can be found in <a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>‘s book <em>Topos Theory</em>, p261.</p> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>This is in stark contrast to the situation for <a class="existingWikiWord" href="/nlab/show/projective+objects">projective objects</a>, which generally do not exist in categories of sheaves.</p> </div> <div class="num_cor"> <h6 id="corollary_3">Corollary</h6> <p>The category of sheaves of modules over any <span class="newWikiWord">sheaf of rings<a href="/nlab/new/sheaf+of+rings">?</a></span> on any <a class="existingWikiWord" href="/nlab/show/small+site">small site</a> also enough injectives.</p> </div> <div class="proof"> <h6 id="proof_11">Proof</h6> <p>Combining prop. <a class="maruku-ref" href="#AbelianSheavesHaveEnoughProjectives"></a> with prop. <a class="maruku-ref" href="#AbHasEnoughInjectives"></a> (which relativizes to any topos).</p> </div> <p>This slick proof of this important fact was pointed out by <a class="existingWikiWord" href="/nlab/show/Colin+McLarty">Colin McLarty</a> in an email to the categories list dated 10 Oct 2010.</p> <h3 id="InjectiveResolutions">Injective resolutions</h3> <div class="num_prop"> <h6 id="proposition_14">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>. Then for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{A}</annotation></semantics></math> there is an <a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, hence a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mo>•</mo></msup><mo>=</mo><mo stretchy="false">[</mo><msup><mi>J</mi> <mn>0</mn></msup><mo>→</mo><mi>⋯</mi><mo>→</mo><msup><mi>J</mi> <mi>n</mi></msup><mo>→</mo><mi>⋯</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>Ch</mi> <mo stretchy="false">(</mo></msub><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> J^\bullet = [J^0 \to \cdots \to J^n \to \cdots] \in Ch_(\mathcal{A}) </annotation></semantics></math></div> <p>equipped with a a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/cochain+complexes">cochain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mo>∼</mo></mover><msup><mi>J</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">X \stackrel{\sim}{\to} J^\bullet</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><mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>J</mi> <mn>0</mn></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>J</mi> <mn>1</mn></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>J</mi> <mi>n</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\to& 0 &\to& \cdots &\to& 0 &\to& \cdots \\ \downarrow && \downarrow && && \downarrow \\ J^0 &\to& J^1 &\to& \cdots &\to& J^n &\to& \cdots } \,. </annotation></semantics></math></div></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+presentation">projective presentation</a>, <a class="existingWikiWord" href="/nlab/show/projective+cover">projective cover</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/projective+module">projective module</a></li> </ul> </li> <li> <p><strong>injective object</strong>, <a class="existingWikiWord" href="/nlab/show/injective+presentation">injective presentation</a>, <a class="existingWikiWord" href="/nlab/show/injective+envelope">injective envelope</a>, <a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/injective+module">injective module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraically+injective+object">algebraically injective object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+object">free object</a>, <a class="existingWikiWord" href="/nlab/show/free+resolution">free resolution</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/free+module">free module</a></li> </ul> </li> <li> <p>flat object, <a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/flat+module">flat module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonality">orthogonality</a></p> </li> </ul> <h2 id="references">References</h2> <p>The notion of <a class="existingWikiWord" href="/nlab/show/injective+modules">injective modules</a> was introduced in</p> <ul> <li id="Baer">R. Baer, <em>Abelian groups that are direct summands of every containing abelian group</em> , Bulletin AMS <strong>46</strong> no. 10 (1940) pp.800-806. (<a href="http://projecteuclid.org/euclid.bams/1183503234">projecteuclid</a>)</li> </ul> <p>The dual notion of <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> was considered explicitly only much later in</p> <ul> <li id="CartanEilenberg56"><a class="existingWikiWord" href="/nlab/show/Henri+Cartan">Henri Cartan</a>, <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <em><a class="existingWikiWord" href="/nlab/show/Homological+Algebra">Homological Algebra</a></em>, Princeton Univ. Press (1956), Princeton Mathematical Series <strong>19</strong> (1999) [<a href="https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19">ISBN:9780691049915</a>, <a href="https://doi.org/10.1515/9781400883844">doi:10.1515/9781400883844</a>, <a href="http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf">pdf</a>]</li> </ul> <p>Textbook accounts:</p> <ul> <li id="HiltonStammbach71"><a class="existingWikiWord" href="/nlab/show/Peter+Hilton">Peter Hilton</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Stammbach">Urs Stammbach</a>, p. 30 in: <em>A course in homological algebra</em>, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4 (<a href="https://link.springer.com/book/10.1007/978-1-4419-8566-8">doi:10.1007/978-1-4419-8566-8</a>, <a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/hilton-stammbach.pdf">pdf</a>)</li> </ul> <p>(used, e.g., for the proof of <a href="#Adjuncts_Injectives">Lemma</a>);</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, sections 9.5, 14.1 of:_<a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a>_</li> </ul> <p>Using tools from the theory of <a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a>, injective objects are discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jiri+Rosicky">Jiri Rosicky</a>, <em>Injectivity and accessible categories</em> (<a href="http://www.math.muni.cz/~rosicky/papers/acc5.ps">ps</a>)</li> </ul> <p><a href="#Baer">Baer’s criterion</a> is discussed in many texts, for example</p> <ul> <li>N. Jacobsen, <em>Basic Algebra II</em>, W.H. Freeman and Company, 1980.</li> </ul> <p>See also</p> <ul> <li> T.-Y. Lam, <em>Lectures on modules and rings</em>, Graduate Texts in Mathematics 189, Springer Verlag (1999).</li> </ul> <p>For injective objects in a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> see</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <em>Handbook of Categorical Algebra vol. 3</em> , Cambridge UP 1994. (section 5.6, pp.314-315)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+Mac+Lane">Saunders Mac Lane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Sheaves in geometry and Logic</em> , Springer Heidelberg 1994. (section IV.10, pp.210-213)</p> </li> <li> <p>D. Higgs, <em>Injectivity in the topos of complete Heyting algebra valued sets</em> , Can. J. Math. <strong>36</strong> (1984) pp.550-568. (<a href="http://cms.math.ca/openaccess/cjm/v36/cjm1984v36.0550-0568.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <a class="existingWikiWord" href="/nlab/show/Fred+Linton">Fred Linton</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Par%C3%A9">Robert Paré</a>, <em>Injective Objects in Topoi II: Connections with the axiom of choice</em> , pp.207-216 in LNM <strong>719</strong> Springer Heidelberg 1979.</p> </li> <li> <p>T. Kenney, <em>Injective Power Objects and the Axiom of Choice</em> , JPAA <strong>215</strong> (2011) pp.131–144.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fred+Linton">Fred Linton</a>, <em>Injective Objects in Topoi III: Stability under coproducts</em> , Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. <strong>29</strong> (1981) pp.341-347.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fred+Linton">Fred Linton</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Par%C3%A9">Robert Paré</a>, <em>Injective Objects in Topoi I: Representing coalgebras as algebras</em> , pp.196-206 in LNM <strong>719</strong> Springer Heidelberg 1979.</p> </li> </ul> <p>Discussion of injective objects (<a class="existingWikiWord" href="/nlab/show/types">types</a>) in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li id="Escardo19"><a class="existingWikiWord" href="/nlab/show/Mart%C3%ADn+Escard%C3%B3">Martín Escardó</a>, <em>Injectives types in univalent mathematics</em> (<a href="https://arxiv.org/abs/1903.01211">arXiv:1903.01211</a>)</li> </ul> <p>For a detailed discussion of internal notions of injectivity see</p> <ul> <li id="Harting">Roswitha Harting, <em>Locally injective abelian groups in a topos</em>, Communications in Algebra 11 (4), 1983.</li> </ul> <p>For injective toposes in the 2-category of <a class="existingWikiWord" href="/nlab/show/bounded+topos">bounded toposes</a> see</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Injective Toposes</em> , pp.284-297 in LNM <strong>871</strong> Springer Heidelberg 1981.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Sketches of an Elephant vol. 2</em> , Cambridge UP 2002. (section C4.3, pp.738-745)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 23, 2024 at 09:28:15. 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