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href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a></p> </li> </ul> <h2 id='sidebar_universal_constructions'>Universal constructions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal construction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>/<a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a>/<a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> </li> </ul> </li> </ul> <h2 id='sidebar_theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+lifting+theorem'>adjoint lifting theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gabriel%E2%80%93Ulmer+duality'>Gabriel-Ulmer duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freyd-Mitchell+embedding+theorem'>Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+type+theory+and+category+theory'>relation between type theory and category theory</a></p> </li> </ul> <h2 id='sidebar_extensions'>Extensions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> </ul> <h2 id='sidebar_applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></li> </ul> <div> <p> <a href='/nlab/edit/category+theory+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='subobjects'>Subobjects</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a><ul><li><a href='#as_classes_of_monomorphisms'>As classes of monomorphisms</a></li><li><a href='#in_terms_of_over_categories'>In terms of over categories</a></li><li><a href='#generalizations'>Generalizations</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#the_poset_of_subobjects'>The poset of subobjects.</a></li><li><a href='#LimitsAndColimits'>Limits and colimits of subobjects</a></li><li><a href='#comparison_with_the_notion_of_subset'>Comparison with the notion of “subset”</a></li></ul></li><li><a href='#Examples'>Examples</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <h3 id='as_classes_of_monomorphisms'>As classes of monomorphisms</h3> <p>A <strong>subobject</strong> of an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> in a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> class of <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphisms</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>a</mi><mo>↪</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'> i: a \hookrightarrow c </annotation></semantics></math></div> <p>into <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math>. (Two morphisms <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>i: a \to c</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>j: b \to c</annotation></semantics></math> are <em>isomorphic</em> if there exists an isomorphism <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>k: a \to b</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>=</mo><mi>j</mi><mi>k</mi></mrow><annotation encoding='application/x-tex'>i = j k</annotation></semantics></math>.)</p> <p>Monos into an object <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/preorder'>preordered</a> by a relation</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>i</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>≤</mo><mo stretchy='false'>(</mo><mi>j</mi><mo>:</mo><mi>b</mi><mo>→</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (i: a \to c) \leq (j: b \to c) </annotation></semantics></math></div> <p>defined by the condition that there exists <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>k: a \to b</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>=</mo><mi>j</mi><mi>k</mi></mrow><annotation encoding='application/x-tex'>i = j k</annotation></semantics></math>. (There is at most one such <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> since <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math> is monic, and such <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> is monic since <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> is monic.)</p> <p>A subobject of <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> may equivalently be defined as an element of the <a class='existingWikiWord' href='/nlab/show/diff/posetal+reflection'>posetal reflection</a> <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sub</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub(c)</annotation></semantics></math> of this preorder.</p> <h3 id='in_terms_of_over_categories'>In terms of over categories</h3> <p>Let <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub></mrow><annotation encoding='application/x-tex'>C_c</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full subcategory</a> of the <a class='existingWikiWord' href='/nlab/show/diff/over+category'>over category</a> <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>/</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>C/c</annotation></semantics></math> on <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'><span><del class='diffmod'> monomorphism</del><ins class='diffmod'> monomorphisms</ins></span></a><span><del class='diffmod'> s.</del><ins class='diffmod'> .</ins> Then</span><math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub></mrow><annotation encoding='application/x-tex'>C_c</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> of subobjects of <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> and the set of isomorphism classes of <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub></mrow><annotation encoding='application/x-tex'>C_c</annotation></semantics></math> is the set of subobjects of <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math>. However this “set” can be in fact a proper class in general, see <a class='existingWikiWord' href='/nlab/show/diff/well-powered+category'>well-powered category</a>.</p> <h3 id='generalizations'>Generalizations</h3> <ul> <li> <p>More generally, in some contexts we may take “subobject” to mean an isomorphism class of morphisms <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>i: a\to c</annotation></semantics></math> satisfying some suitable condition other than being a monomorphism (usually a stronger one). Common choices are <a class='existingWikiWord' href='/nlab/show/diff/strong+monomorphism'>strong monomorphisms</a>, <a class='existingWikiWord' href='/nlab/show/diff/regular+monomorphism'>regular monomorphisms</a>, or the right class of some <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+factorization+system'>orthogonal factorization system</a>. (The latter choice has the advantage that then <a class='existingWikiWord' href='/nlab/show/diff/image'>images</a> will automatically exist.)</p> <ul> <li>For example, in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> a monomorphism is just a continuous injective function, whereas the strong and regular monomorphisms coincide and are the subspace embeddings. In some contexts at least, one can argue that subspace embeddings are a more appropriate notion of “subobject” in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> (for example, if one wants to exhibit it as a <a class='existingWikiWord' href='/nlab/show/diff/locally+bounded+category'>locally bounded category</a>). A similar thing happens in a <a class='existingWikiWord' href='/nlab/show/diff/quasitopos'>quasitopos</a>.</li> </ul> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>partial order</a> on the collection of subobjects internalizes into contexts more general than <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>. For instance in every <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a> the <a class='existingWikiWord' href='/nlab/show/diff/subobject+classifier'>subobject classifier</a> <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ω</mi></mrow><annotation encoding='application/x-tex'>\Omega</annotation></semantics></math> has the structure of an internal poset (see there).</p> </li> </ul> <h2 id='properties'>Properties</h2> <h3 id='the_poset_of_subobjects'>The poset of subobjects.</h3> <ul> <li> <p>For <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/accessible+category'>accessible category</a>, <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>c \in C</annotation></semantics></math> any object, the poset <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sub</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub(c)</annotation></semantics></math> of subobjects of <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/small+category'>small category</a>.</p> </li> <li> <p>Suppose <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/well-powered+category'>well-powered category</a>. Denote by <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sub</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub(X)</annotation></semantics></math> the poset of subobjects of object <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>. The correspondence <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sub</mi><mo>:</mo><mi>X</mi><mo>↦</mo><mi>Sub</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub:X\mapsto Sub(X)</annotation></semantics></math> may be extended to a <a class='existingWikiWord' href='/nlab/show/diff/contravariant+functor'>contravariant functor</a> <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>Pos</mi></mrow><annotation encoding='application/x-tex'>E \to Pos</annotation></semantics></math> (that is a functor <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>E</mi> <mi>op</mi></msup><mo>→</mo><mi>Pos</mi></mrow><annotation encoding='application/x-tex'>E^op \to Pos</annotation></semantics></math>), namely if <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_39' xmlns='http://www.w3.org/1998/Math/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 arbitrary and <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi><mo>:</mo><mi>S</mi><mo>↪</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>m:S\hookrightarrow Y</annotation></semantics></math> is an element of <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sub</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub(Y)</annotation></semantics></math>, i.e. monic, then the pullback <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>m</mi><mo stretchy='false'>)</mo><mo>:</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>f^*(m):f^*(S)\to X</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi></mrow><annotation encoding='application/x-tex'>m</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is automatically a monic; the correspondence <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi><mo>↦</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>m</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>m\mapsto f^*(m)</annotation></semantics></math> describes <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sub</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub(f)</annotation></semantics></math> at the level of representatives of subobjects.</p> </li> </ul> <h3 id='LimitsAndColimits'>Limits and colimits of subobjects</h3> <p>Assume that the ambient category has all <a class='existingWikiWord' href='/nlab/show/diff/limit'><span><del class='diffmod'> limit</del><ins class='diffmod'> limits</ins></span></a><span><del class='diffmod'> s</del><ins class='diffmod'> </ins> and</span><a class='existingWikiWord' href='/nlab/show/diff/colimit'><span><del class='diffmod'> colimit</del><ins class='diffmod'> colimits</ins></span></a><span><del class='diffmod'> s</del><ins class='diffmod'> </ins> considered in the following.</span></p> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>X \in C</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a>, <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sub</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub(X)</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> of subobjects and<br /><math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U_1, U_2 \hookrightarrow X</annotation></semantics></math> two <a class='existingWikiWord' href='/nlab/show/diff/subobject'><span><del class='diffmod'> subobject</del><ins class='diffmod'> subobjects</ins></span></a><span><del class='diffmod'> s,</del><ins class='diffmod'> ,</ins></span></p> <ul> <li> <p>their <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sub</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub(X)</annotation></semantics></math> is denoted <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>U_1 \cap U_2</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>U_1 \wedge U_2</annotation></semantics></math> and called the <strong><a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersection</a></strong> or <strong><a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a></strong> of the two subobjects;</p> </li> <li> <p>their <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sub</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub(X)</annotation></semantics></math> is denoted <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∪</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>U_1 \cup U_2</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>U_1 \vee U_2</annotation></semantics></math> and called the <strong><a class='existingWikiWord' href='/nlab/show/diff/union'>union</a></strong> or <strong><a class='existingWikiWord' href='/nlab/show/diff/join'>join</a></strong> of the two subobjects.</p> </li> </ul> <p>Two subobjects <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>Sub</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>U_1, U_2 \in Sub_C(X)</annotation></semantics></math> are called <strong>disjoint</strong> if <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>U_1 \cap U_2 = \emptyset</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a>.</p> </div> <div class='num_prop' id='IntersectionAndUnionAsFiberProductAndCoimage'> <h6 id='proposition'>Proposition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a>.</p> <ol> <li> <p>The intersection of two subobjects in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Sub</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub_C(X)</annotation></semantics></math> is their <a class='existingWikiWord' href='/nlab/show/diff/pullback'>fiber product</a> in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>: the <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>U</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ U_1 \cap U_2 &\to& U_2 \\ \downarrow && \downarrow \\ U_1 &\to& X } </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> diagram.</p> </li> <li> <p>The union of two subobjects <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>Sub</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>U_1, U_2 \in Sub_C(X)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><msub><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow></msub><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>U_1 \coprod_{U_1 \cap U_2} U_2</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>: the <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>U</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>U</mi> <mn>1</mn></msub><mo>∪</mo><msub><mi>U</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ U_1 \cap U_2 &\to& U_2 \\ \downarrow && \downarrow \\ U_1 &\to& U_1 \cup U_2 } </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> diagram.</p> </li> <li> <p>This last pushout diagram is also a <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> diagram.</p> </li> </ol> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>For the first point: Since <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphism</a> are (as discussed there) stable under <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> and composition, the fiber product is a subobject. Its <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> as a <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a> in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> then implies its universal property as a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Sub</mi> <mi>C</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sub_C(X)</annotation></semantics></math>.</p> <p>For the second point: by the same kind of argument, it is sufficient to show that the canonical morphism <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><msub><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow></msub><msub><mi>U</mi> <mn>2</mn></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U_1 \coprod_{U_1 \cap U_2} U_2 \to X</annotation></semantics></math> exhibits the coproduct as a subobject.</p> <p>Since <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'><span><del class='diffmod'> monomorphism</del><ins class='diffmod'> monomorphisms</ins></span></a><span><del class='diffmod'> s</del><ins class='diffmod'> </ins> (as discussed there) are characterized by the fact that the</span><a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> along themselves is their domain, it is sufficient to show that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>U</mi> <mn>1</mn></msub><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow></munder><msub><mi>U</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>→</mo><mi>Id</mi></mover></mtd> <mtd><msub><mi>U</mi> <mn>1</mn></msub><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow></munder><msub><mi>U</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mi>id</mi></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mn>1</mn></msub><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow></munder><msub><mi>U</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>→</mo><mi>i</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ U_1 \coprod_{U_1 \cap U_2} U_2 &\stackrel{Id}{\to}& U_1 \coprod_{U_1 \cap U_2} U_2 \\ {}^{\mathllap{id}}\downarrow && \downarrow \\ U_1 \coprod_{U_1 \cap U_2} U_2 &\stackrel{i}{\to}& X } </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> diagram. For showing this we use that in a <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a> we have <a class='existingWikiWord' href='/nlab/show/diff/pullback-stable+colimit'>universal colimits</a>, so that equivalently it is sufficient to show that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow></munder><msub><mi>U</mi> <mn>2</mn></msub><mo>≃</mo><mo stretchy='false'>(</mo><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msup><mi>i</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow></munder><mo stretchy='false'>(</mo><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> U_1 \coprod_{U_1 \cap U_2} U_2 \simeq (i^* U_1) \coprod_{i^* (U_1 \cap U_2)} (i^* U_2) \,. </annotation></semantics></math></div> <p>To see this, again use <a class='existingWikiWord' href='/nlab/show/diff/pullback-stable+colimit'>universal colimits</a> to get</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>U</mi> <mn>1</mn></msub></mtd> <mtd><mo>≃</mo><msub><mi>U</mi> <mn>1</mn></msub><msub><mo>×</mo> <mi>X</mi></msub><mo stretchy='false'>(</mo><msub><mi>U</mi> <mn>1</mn></msub><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow></munder><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><mo stretchy='false'>(</mo><msub><mi>U</mi> <mn>1</mn></msub><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><msub><mo>×</mo> <mi>X</mi></msub><mo stretchy='false'>(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow></munder><mo stretchy='false'>(</mo><msub><mi>U</mi> <mn>1</mn></msub><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mi>U</mi> <mn>1</mn></msub><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow></munder><mo stretchy='false'>(</mo><msub><mi>U</mi> <mn>1</mn></msub><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mi>U</mi> <mn>1</mn></msub><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow></munder><mo stretchy='false'>(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msub><mi>U</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} i^* U_1 & \simeq U_1 \times_X (U_1 \coprod_{U_1 \cap U_2} U_2) \\ & \simeq (U_1 \times_X U_1) \coprod_{U_1 \times_X (U_1 \cap U_2)} (U_1 \times_X U_2) \\ & \simeq U_1 \coprod_{U_1 \cap U_2} (U_1 \times_X U_2) \\ & \simeq U_1 \coprod_{U_1 \cap U_2} (U_1 \cap U_2) \\ & \simeq U_1 \end{aligned} </annotation></semantics></math></div> <p>and similarly</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>U</mi> <mn>2</mn></msub><mo>≃</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'> i^* U_2 \simeq U_2 </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>≃</mo><mo stretchy='false'>(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> i^* (U_1 \cap U_2) \simeq (U_1 \cap U_2) \,. </annotation></semantics></math></div> <p>This proves the second point.</p> <p>The third point is directly verified by checking the universal property.</p> </div> <h3 id='comparison_with_the_notion_of_subset'>Comparison with the notion of “subset”</h3> <p>The notion of subobject figures prominently in <a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>topos theory</a> and in other approaches to <a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set theory</a> based on categories. It is not an exact translation of the usual notion of “subset” in traditional set theory; in ordinary set theory, the notion of subset is defined in terms of a global elementhood relation between sets, which one doesn’t have in categorical set theory (and which one wouldn’t necessarily want: it violates the <a class='existingWikiWord' href='/nlab/show/diff/principle+of+equivalence'>principle of equivalence</a> in the sense of not being invariant with respect to <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>).</p> <p>Category-theoretically, the traditional notion of subset gives a way of picking out a canonical representative or “normal form” among all the monos in an isomorphism class. As we intimated, there is no intrinsic way of defining such representatives in the theory of toposes: such would have to be considered an extra structure on a topos. Mathematically, there is no particular gain in having such structure around; at best it enables a traditional mode of discourse in which subsets are concrete maps, and to this end it can function as a linguistic or psychological convenience.</p> <p>On the other hand, there is no particular harm either in having such structure around, as long as one remembers that it is not an isomorphism invariant. People will instinctively turn to canonical representatives whenever they can – think of what we would tell a student who asks for help understanding how to multiply elements in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℤ</mi> <mn>13</mn></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_13</annotation></semantics></math> – and even category theorists do so when they are available.</p> <h2 id='Examples'>Examples</h2> <ul> <li> <p>A subobject in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> is a <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a>.</p> </li> <li> <p>A subobject in <a class='existingWikiWord' href='/nlab/show/diff/Grp'>Grp</a> is a <a class='existingWikiWord' href='/nlab/show/diff/subgroup'>subgroup</a>.</p> </li> <li> <p>A subobject in <a class='existingWikiWord' href='/nlab/show/diff/Ring'>Ring</a> is a <a class='existingWikiWord' href='/nlab/show/diff/subring'>subring</a>.</p> </li> <li> <p>A subobject in <math class='maruku-mathml' display='inline' id='mathml_b0fdd78a64c72de465ee8776c9c6890911ccc3f5_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/Mod'>Mod</a> is a <a class='existingWikiWord' href='/nlab/show/diff/submodule'>submodule</a>.</p> </li> <li> <p>A subobject of a <a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a> is a <a class='existingWikiWord' href='/nlab/show/diff/subrepresentation'>subrepresentation</a>.</p> </li> <li> <p>A subobject (a <a class='existingWikiWord' href='/nlab/show/diff/subfunctor'>subfunctor</a>) of a <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a> is a <a class='existingWikiWord' href='/nlab/show/diff/sieve'>sieve</a>.</p> </li> <li> <p>A subobject of a <a class='existingWikiWord' href='/nlab/show/diff/bundle'>bundle</a> (hence in a <a class='existingWikiWord' href='/nlab/show/diff/over+category'>slice category</a>) is a <a class='existingWikiWord' href='/nlab/show/diff/sub-bundle'>sub-bundle</a>.</p> </li> </ul> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><strong>subobject</strong></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pure+subobject'>pure subobject</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/improper+subset'>improper subobject</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subobject+in+an+%28infinity%2C1%29-category'>subobject in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/property+sup'>property sup</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subtype'>subtype</a></p> </li> </ul> <h2 id='references'>References</h2> <p>Standard textbook references include section I.3 of</p> <ul id='MacLaneMoerdijk'> <li><a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ieke+Moerdijk'>Ieke Moerdijk</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Sheaves+in+Geometry+and+Logic'>Sheaves in Geometry and Logic</a></em></li> </ul> <p>and</p> <ul id='Johnstone'> <li><a class='existingWikiWord' href='/nlab/show/diff/Peter+Johnstone'>Peter Johnstone</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Sketches+of+an+Elephant'>Sketches of an Elephant</a></em></li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on December 11, 2023 at 21:03:21. 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