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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/7743/#Item_28" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry is about the basic notion of “monic” in relation to <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. For the notion of “monic” in relation to <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a> in <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a>, see at <em><a class="existingWikiWord" href="/nlab/show/monic+polynomial">monic polynomial</a></em>.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ExamplesOfMonosThatAreEpiButNotIso'>Examples of monos that are epi but not iso</a></li> </ul> <li><a href='#properties'>Properties</a></li> <li><a href='#variations'>Variations</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <em>monomorphism</em> is the generalization of the notion of <em>injective map of sets</em> from the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> to arbitrary <a class="existingWikiWord" href="/nlab/show/category">categories</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/formal+duality">formally dual</a> concept is that of <em><a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a></em>, which similarly generalizes (or strengthens) the concept of <a class="existingWikiWord" href="/nlab/show/surjective+function">surjective function</a>.</p> <p>Common jargon includes “is a mono” or “is monic” for “is a monomorphism”, and “is an epi” or “is epic” for “is an epimorphism”, and “is an iso” for “is an isomorphism”.</p> <h2 id="definition">Definition</h2> <p>A <a class="existingWikiWord" href="/nlab/show/morphism">morphism</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>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> in some <a class="existingWikiWord" href="/nlab/show/category">category</a> is called a <em>monomorphism</em> (sometimes abbrieviated to <em>mono</em>, or described as being <em>monic</em>), if for every <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>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> and every <a class="existingWikiWord" href="/nlab/show/pair+of+parallel+morphisms">pair of parallel morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">g_1,g_2 \colon Z \to X</annotation></semantics></math> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>f</mi><mo>∘</mo><msub><mi>g</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mi>f</mi><mo>∘</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msub><mi>g</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mi>g</mi> <mn>2</mn></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( f \circ g_1 \,=\, f \circ g_2 \right) \;\Rightarrow \; \left( g_1 \,=\, g_2 \right) \,. </annotation></semantics></math></div> <p>Stated more abstractly, this says 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 monomorphism if for every <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">Z</annotation></semantics></math> 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><mi>Hom</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(Z,-)</annotation></semantics></math> takes it to an <a class="existingWikiWord" href="/nlab/show/injective+function">injective function</a> between <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>f</mi> <mo>*</mo></msub><mphantom><mi>AA</mi></mphantom></mrow></mover><mi>Hom</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom(Z,X) \overset{\phantom{AA} f_\ast \phantom{AA}}{\hookrightarrow} Hom(Z,Y) \,. </annotation></semantics></math></div> <p>Since injective functions are precisely the monomorphisms in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> (example <a class="maruku-ref" href="#MonomorphismsInSet"></a> below) this may be stated as saying 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 monomorphism if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>Z</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(Z,f)</annotation></semantics></math> is a monomorphism for all objects <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">Z</annotation></semantics></math>.</p> <p>Finally, <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> being a monomorphism 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">\mathcal{C}</annotation></semantics></math> means equivalently that it is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{op}</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <h3 id="general">General</h3> <div class="num_example"> <h6 id="example">Example</h6> <p><strong>(monomorphisms in preorders)</strong></p> <p>In a <a class="existingWikiWord" href="/nlab/show/preorder">preorder</a>, all arrows are mono because they satisfy the required condition vacuously (any pair of parallel arrows is equal in a preorder).</p> </div> <div class="num_example" id="MonomorphismsInSet"> <h6 id="example_2">Example</h6> <p><strong>(monomorphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>)</strong></p> <p>The monomorphisms in the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> and <a class="existingWikiWord" href="/nlab/show/functions">functions</a> between them are precisely the <a class="existingWikiWord" href="/nlab/show/injective+functions">injective functions</a>.</p> </div> <div class="num_example" id="MonomorphismsInCat"> <h6 id="example_3">Example</h6> <p><strong>(monomorphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>)</strong></p> <p>The monomorphisms in the category <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> of <a class="existingWikiWord" href="/nlab/show/categories">categories</a> and <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between them are precisely the <a class="existingWikiWord" href="/nlab/show/embeddings+of+categories">embeddings of categories</a>, i.e. the <a class="existingWikiWord" href="/nlab/show/injective-on-objects+functor">injective-on-objects</a> <a class="existingWikiWord" href="/nlab/show/faithful+functors">faithful functors</a>.</p> </div> <div class="num_example"> <h6 id="example_4">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> is both a monomorphism and an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>.</p> </div> <p>But beware that the converse fails:</p> <h3 id="ExamplesOfMonosThatAreEpiButNotIso">Examples of monos that are epi but not iso</h3> <p>The following lists some examples of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> that are both <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> and <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a>, but not necessarily <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>.</p> <p> <div class='num_remark'> <h6>Example</h6> <p></p> <p>In the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+spaces">Hausdorff topological spaces</a>, the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/dense+subspace">dense subspace</a> is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>.</p> <p></p> </div> </p> <p>See <a href="Hausdorff+space#DenseSubspaceInclusionsAreEpimorphismsAmongHausdorffSpaces">this Prop.</a> for proof.</p> <p> <div class='num_remark'> <h6>Example</h6> <p></p> <p>In <a class="existingWikiWord" href="/nlab/show/unital+ring">unital</a> <a class="existingWikiWord" href="/nlab/show/Rings">Rings</a>, the canonical inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mover><mo>↪</mo><mi>i</mi></mover><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \overset{i}{\hookrightarrow} \mathbb{Q}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a> into the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>.</p> <p></p> </div> </p> <p>See <a href="Ring#InclusionOfIntegersIntoRationalsIsEpimorphismOfRings">this Prop.</a> for proof.</p> <h2 id="properties">Properties</h2> <p>We list the following properties without their (easy) proofs. The proofs can be found spelled out (dually) at <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>.</p> <p> <div class='num_prop' id='BasicCharacterizationOfMonomorphisms'> <h6>Proposition</h6> <p>The following are equivalent:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>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> is a monomorphism 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> </li> <li> <p><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 an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math>;</p> </li> <li> <p>postcomposition with <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 <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>: that is, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>:</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \circ -: Hom(c,x) \to Hom(c,y)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/injection">injection</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</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><mover><mo>→</mo><mi>Id</mi></mover></mtd> <mtd><mi>x</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mi>Id</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><munder><mo>→</mo><mi>f</mi></munder></mtd> <mtd><mi>y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ x &\stackrel{Id}{\to}& x \\ {}^{Id}\downarrow && \downarrow^{f} \\ x &\underset{f}{\to}& y } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram.</p> </li> </ul> <p></p> </div> </p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f \colon x \to y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>y</mi><mo>→</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">g \colon y \to z</annotation></semantics></math> are monomorphisms, so is their composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">g f</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">g f</annotation></semantics></math> is an monomorphism, so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>→</mo><mi>x</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>y</mi></mrow><annotation encoding="application/x-tex"> t \to x \stackrel{\longrightarrow}{\longrightarrow} y </annotation></semantics></math></div> <p>is a monomorphism.</p> </div> <p>The converse of the above proposition fails, and a mononomorphism that is the equalizer of some pair of morphisms is called a <a class="existingWikiWord" href="/nlab/show/regular+monomorphism">regular monomorphism</a>.</p> <div class="num_prop" id="MonomorphismsArePreservedByPullback"> <h6 id="proposition_3">Proposition</h6> <p>Monomorphisms are preserved by <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>.</p> </div> <p>(In an <a class="existingWikiWord" href="/nlab/show/adhesive+category">adhesive category</a> they are also preserved by <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>.)</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Monomorphisms are preserved by any <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a>, or more generally any functors that preserves pullbacks.</p> </div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>Monomorphisms are <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected</a> by <a class="existingWikiWord" href="/nlab/show/faithful+functors">faithful functors</a>.</p> </div> <p>We have seen some ways in which monomorphisms get along with limits. Here is another:</p> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>Any morphism from a terminal object is a monomorphism. The product of monomorphisms is a monomorphism.</p> </div> <p>Monomorphisms do not get along quite as well with colimits. For example, the unique morphism from the initial object is not always an monomorphism, and the canonical morphisms from the summands into a coproduct, e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>:</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">i_1 : x_1 \to x_1 + x_2</annotation></semantics></math>, are not always monomorphisms (though these results do hold in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>). However, the unique morphism from the initial object is a monomorphism when the initial object is <a class="existingWikiWord" href="/nlab/show/strict+initial+object">strict</a>, and the canonical morphisms into a coproduct are monomorphisms when the coproduct is a <a class="existingWikiWord" href="/nlab/show/disjoint+coproduct">disjoint coproduct</a>.</p> <h2 id="variations">Variations</h2> <p>At <a href="/nlab/show/epimorphism#Variations">epimorphism</a> there is a long list of variations on the concept of epimorphism. Each of these, of course, has a dual notion for monomorphism, but the ones most commonly used are:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/split+monomorphism">split monomorphism</a> = morphism which has a <a class="existingWikiWord" href="/nlab/show/retraction">retraction</a></li> <li><a class="existingWikiWord" href="/nlab/show/normal+monomorphism">normal monomorphism</a> = a <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of some morphism (in a category with <a class="existingWikiWord" href="/nlab/show/zero+morphisms">zero morphisms</a>)</li> <li><a class="existingWikiWord" href="/nlab/show/regular+monomorphism">regular monomorphism</a> = an <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> of some <a class="existingWikiWord" href="/nlab/show/parallel+pair">parallel pair</a></li> <li><a class="existingWikiWord" href="/nlab/show/strong+monomorphism">strong monomorphism</a> = a monomorphism <a class="existingWikiWord" href="/nlab/show/orthogonality">right orthogonal</a> to <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a></li> <li>monomorphism</li> </ul> <p>Frequently, regular and strong monos coincide. For instance, this is the case in any <a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a>, and also in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, where they are the subspace inclusions (the plain monomorphisms are the injective functions).</p> <p>Sometimes, all monomorphisms are regular—this seems to happen a bit more frequently than for epimorphisms. For instance, this is the case in any <a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a> (including any <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, such as <a class="existingWikiWord" href="/nlab/show/Set">Set</a>), but also in any <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, and also in the category <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> and in any <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, all monomorphisms are normal. But this is not so in <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a>, where (despite the fact that all monomorphisms are regular), the normal monos are the inclusions of <a class="existingWikiWord" href="/nlab/show/normal+subgroups">normal subgroups</a> (hence the name). In any <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a>, all regular monos are normal, but not all monos need be regular.</p> <p>In a <a class="existingWikiWord" href="/nlab/show/Boolean+topos">Boolean topos</a>, such as <a class="existingWikiWord" href="/nlab/show/Set">Set</a> (in <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a>), any monomorphism with <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a> domain is split. Of course, no mono with empty domain and inhabited codomain can be split (in contrast to the dual case, where it can happen that all epimorphisms split – the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> classes of monomorphism define <a class="existingWikiWord" href="/nlab/show/subobject">subobjects</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monomorphism+in+an+%28%E2%88%9E%2C1%29-category">monomorphism in an (∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/n-monomorphism">n-monomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/image">image</a>, <a class="existingWikiWord" href="/nlab/show/coimage">coimage</a></p> </li> <li> <p>a category in which all morphisms are monomorphisms is called a <em><a class="existingWikiWord" href="/nlab/show/left+cancellative+category">left cancellative category</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jointly+monic+morphisms">jointly monic morphisms</a></p> </li> </ul> <h2 id="references">References</h2> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, §I.5 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (1971, second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Section 1.7 in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em> Vol. 1: <em>Basic Category Theory</em> [<a href="https://doi.org/10.1017/CBO9780511525858">doi:10.1017/CBO9780511525858</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 25, 2024 at 06:24:24. 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