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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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>Structured sets</title></head> <body> <h1 id="structured_sets">Structured sets</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#abstract'>Abstract</a></li> <li><a href='#Concrete'>Concrete</a></li> </ul> <li><a href='#conversions'>Conversions</a></li> <ul> <li><a href='#abstract_to_concrete'>Abstract to concrete</a></li> <li><a href='#concrete_to_abstract'>Concrete to abstract</a></li> <li><a href='#back_and_forth'>Back and forth</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#structured_objects'>Structured objects</a></li> <li><a href='#see_also'>See also</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>structured set</em> is, of course, a <a class="existingWikiWord" href="/nlab/show/set">set</a> equipped with <a class="existingWikiWord" href="/nlab/show/extra+structure">extra</a> <a class="existingWikiWord" href="/nlab/show/structure">structure</a>. It is not the individual structured set that matters so much as the <a class="existingWikiWord" href="/nlab/show/concrete+category">concrete</a> <a class="existingWikiWord" href="/nlab/show/category">category</a> of sets with a particular sort of structure.</p> <h2 id="definitions">Definitions</h2> <h3 id="abstract">Abstract</h3> <p>Very abstractly, we may define a <strong>structured set</strong> as an <a class="existingWikiWord" href="/nlab/show/object">object</a> of any <a class="existingWikiWord" href="/nlab/show/concrete+category">concrete category</a>, that is an object of any 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> equipped with a <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful 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>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U\colon C \to Set</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/category+of+sets">category of sets</a>. (Some authors require that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> for a concrete category, but we do not need that here.) Given two structured sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> (in the same 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>), a <a class="existingWikiWord" href="/nlab/show/function">function</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>U</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f\colon U(X) \to U(Y)</annotation></semantics></math> between their underlying sets <strong>preserves</strong> the structure if it lies in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>, that is if there exists a (necessarily unique since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is faithful) <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><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\tilde{f}\colon X \to Y</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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">U(\tilde{f}) = f</annotation></semantics></math>.</p> <h3 id="Concrete">Concrete</h3> <p>More concretely, we may define a <strong>type of structure on sets</strong> as an operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> that, to any set <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>, assigns a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(A)</annotation></semantics></math> of <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structures</strong> on <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>. At minimum, we should have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>T</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(A) \cong T(B)</annotation></semantics></math> whenever <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 \cong B</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≅</mo></mrow><annotation encoding="application/x-tex">\cong</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> 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>), so that the concept is <a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural</a>. But for good behaviour, we actually want something more coherent; we want an additional operation that, to any <a class="existingWikiWord" href="/nlab/show/bijection">bijection</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>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon A \to B</annotation></semantics></math>, assigns a bijection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>T</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(f)\colon T(A) \to T(B)</annotation></semantics></math>, such that:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">T(\id_A) = \id_{T(A)}</annotation></semantics></math>,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mi>T</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(f g) = T(f) T(g)</annotation></semantics></math>.</li> </ul> <p>In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Set</mi> <mo>≅</mo></msub><mo>→</mo><msub><mi>Set</mi> <mo>≅</mo></msub></mrow><annotation encoding="application/x-tex">T\colon Set_\cong \to Set_\cong</annotation></semantics></math> (or equivalently <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Set</mi> <mo>≅</mo></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">T\colon Set_\cong \to Set</annotation></semantics></math>) is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from the <a class="existingWikiWord" href="/nlab/show/underlying+groupoid">underlying groupoid</a> of <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> to itself (or equivalently to all of <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>). In particular, any <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> of a single set <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> defines an automorphism of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structures on <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>, giving an <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">S_A</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(A)</annotation></semantics></math>.</p> <p>(Compare the notion of <a class="existingWikiWord" href="/nlab/show/structure+type">structure type</a> from <a class="existingWikiWord" href="/nlab/show/combinatorics">combinatorics</a>, which is a set-valued functor on the groupoid of <a class="existingWikiWord" href="/nlab/show/finite+sets">finite sets</a>. Every combinatorial structure type can be interpreted as a type of structure, where only finite sets are capable of supporting the structure.)</p> <p>Given a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> of structure on sets, we define a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structured set</strong> to be a set <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> equipped with an <a class="existingWikiWord" href="/nlab/show/element">element</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(A)</annotation></semantics></math>. Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structured sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = (A,\sigma)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y = (B,\tau)</annotation></semantics></math>, a bijection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon A \to B</annotation></semantics></math> <strong>preserves</strong> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>τ</mi></mrow><annotation encoding="application/x-tex">T(f)(\sigma) = \tau</annotation></semantics></math>.</p> <p>In general, there is no notion of whether an arbitrary function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon A \to B</annotation></semantics></math> preserves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structure, although such a notion may be defined in many cases. So to get a concrete construction of a <a class="existingWikiWord" href="/nlab/show/concrete+category">concrete category</a>, we specify whatever morphisms we like, subject to the restriction that they form a category and have the correct <a class="existingWikiWord" href="/nlab/show/core">core</a> given above.</p> <p><a class="existingWikiWord" href="/nlab/show/Bourbaki">Bourbaki</a>'s theory of structure, while not described in category-theoretic terms, is essentially the above.</p> <h2 id="conversions">Conversions</h2> <p>Morally, either of the abstract and concrete versions can be converted into the other. Technically, there are some restrictions.</p> <h3 id="abstract_to_concrete">Abstract to concrete</h3> <p>Given 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> and a faithful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U\colon C \to Set</annotation></semantics></math>, we may define a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> of structure on sets as follows:</p> <p>For each set <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>, consider the <a class="existingWikiWord" href="/nlab/show/essential+fiber">essential fibre</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> over <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>, the collection of pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,f)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f\colon U(X) \to A</annotation></semantics></math> is a bijection. We consider two such pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\sigma)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,\tau)</annotation></semantics></math> to be <a class="existingWikiWord" href="/nlab/show/equivalent">equivalent</a> if there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">h\colon X \to Y</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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>;</mo><mi>τ</mi><mo>=</mo><mi>σ</mi></mrow><annotation encoding="application/x-tex">U(h) ; \tau = \sigma</annotation></semantics></math>. (Because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is faithful, any such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> must be unique.) Define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(A)</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a> of the essential fibre modulo this <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a>. That is, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structure on <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 equivalence class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[(X,\sigma)]</annotation></semantics></math>.</p> <p>Given a bijection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon A \to B</annotation></semantics></math>, we must define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>T</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(f)\colon T(A) \to T(B)</annotation></semantics></math>. So given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\sigma)</annotation></semantics></math> as above, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(f)</annotation></semantics></math> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[(X,\sigma)]</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>σ</mi><mo>;</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[(X,\sigma;f)]</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(B)</annotation></semantics></math>. It's easy to check that this is well defined as a function from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(A)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(B)</annotation></semantics></math>; we can also check that this makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> into a functor and that the abstract and concrete definitions of whether a bijection preserves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structure agree.</p> <p>Technicality: 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/large+category">large category</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(A)</annotation></semantics></math> might be a <a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a> instead of a set. In this case, we can pass to a larger <a class="existingWikiWord" href="/nlab/show/universe">universe</a>; it is not essential for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Set</mi> <mo>≅</mo></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">T\colon Set_\cong \to Set</annotation></semantics></math> that both copies of <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> be the same size. But for the above description to make sense as it is, we must require that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> have essentially small fibres.</p> <h3 id="concrete_to_abstract">Concrete to abstract</h3> <p>Given a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> of structure on sets, we cannot quite reconstruct the 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>, but we can reconstruct its <a class="existingWikiWord" href="/nlab/show/core">core</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>≅</mo></msub></mrow><annotation encoding="application/x-tex">C_\cong</annotation></semantics></math>. That is, we can say what the <a class="existingWikiWord" href="/nlab/show/objects">objects</a> and <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> are, if not the <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in general.</p> <p>An object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is simply a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\sigma)</annotation></semantics></math> consisting of a set <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> and an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(A)</annotation></semantics></math>. Given two such objects, an isomorphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\sigma)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,\tau)</annotation></semantics></math> is simply a structure-preserving map from <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> to <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">B</annotation></semantics></math>, that is a bijection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon A \to B</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>τ</mi></mrow><annotation encoding="application/x-tex">T(f)(\sigma) = \tau</annotation></semantics></math>. Then it is straightforward to check that this defines a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>≅</mo></msub></mrow><annotation encoding="application/x-tex">C_\cong</annotation></semantics></math>. This groupoid, the <strong>groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structured sets</strong>, is naturally equipped with a faithful <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><msub><mi>C</mi> <mo>≅</mo></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U\colon C_\cong \to Set</annotation></semantics></math>, given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">U(A,\sigma) \coloneqq A</annotation></semantics></math>.</p> <p>While defining isomorphisms of structured sets is an exact science, choosing more general morphisms of structured sets is something of an art. In principle, we may define a (<em>not</em> the!) <strong>category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structured sets</strong> by picking, for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\sigma)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,\tau)</annotation></semantics></math>, a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>σ</mi><mo>,</mo><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{\sigma,\tau}(A,B)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/functions">functions</a> from <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> to <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">B</annotation></semantics></math>, such that:</p> <ul> <li> <p>whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mi>Hom</mi> <mrow><mi>σ</mi><mo>,</mo><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in Hom_{\sigma,\tau}(A,B)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><msub><mi>Hom</mi> <mrow><mi>τ</mi><mo>,</mo><mi>υ</mi></mrow></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g \in Hom_{\tau,\upsilon}(B,C)</annotation></semantics></math>, then <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><msub><mi>Hom</mi> <mrow><mi>σ</mi><mo>,</mo><mi>υ</mi></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f ; g \in Hom_{\sigma,\upsilon}(A,C)</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/identity+map">identity map</a> on <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> belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mi>σ</mi><mo>,</mo><mi>σ</mi></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{\sigma,\sigma}(A,A)</annotation></semantics></math>; and</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> <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> from <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> to <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">B</annotation></semantics></math> is an isomorphism (as defined above) if and only if both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mi>Hom</mi> <mrow><mi>σ</mi><mo>,</mo><mi>τ</mi></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in Hom_{\sigma,\tau}(A,B)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∈</mo><msub><mi>Hom</mi> <mrow><mi>τ</mi><mo>,</mo><mi>σ</mi></mrow></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1} \in Hom_{\tau,\sigma}(B,A)</annotation></semantics></math>.</p> </li> </ul> <p>The last condition states precisely that the <a class="existingWikiWord" href="/nlab/show/underlying+groupoid">underlying groupoid</a> of any concrete category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structured sets is the groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structured sets.</p> <p>Any category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-structured sets is still (like the groupoid of such sets) a concrete category.</p> <h3 id="back_and_forth">Back and forth</h3> <p>If we start with a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> of structures on sets, construct from this a groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>≅</mo></msub></mrow><annotation encoding="application/x-tex">C_\cong</annotation></semantics></math> and a faithful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><msub><mi>C</mi> <mo>≅</mo></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U\colon C_\cong \to Set</annotation></semantics></math> and then construct from this another type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">T'</annotation></semantics></math> of structures, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">T'</annotation></semantics></math> will be equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in the sense that there is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> between them as functors from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Set</mi> <mo>≅</mo></msub></mrow><annotation encoding="application/x-tex">Set_\cong</annotation></semantics></math> to <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>.</p> <p>If we start with 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> and a faithful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U\colon C \to Set</annotation></semantics></math>, construct from this a type of structures <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Set</mi> <mo>≅</mo></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">T\colon Set_\cong \to Set</annotation></semantics></math>, and then construct from this a groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>≅</mo></msub></mrow><annotation encoding="application/x-tex">C_\cong</annotation></semantics></math> with a faithful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><msub><mi>C</mi> <mo>≅</mo></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U\colon C_\cong \to Set</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>≅</mo></msub></mrow><annotation encoding="application/x-tex">C_\cong</annotation></semantics></math> will in fact be the <a class="existingWikiWord" href="/nlab/show/core">core</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><msub><mi>C</mi> <mo>≅</mo></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U\colon C_\cong \to Set</annotation></semantics></math> to restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U\colon C \to Set</annotation></semantics></math> to this core, up to <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>.</p> <p>(Do we need proofs?)</p> <p>Thus the abstract and concrete approaches to structured sets are equivalent, except that the concrete approach does not include a specification of what are the noninvertible morphisms between structured sets.</p> <h2 id="examples">Examples</h2> <p>Almost everything in contemporary <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> is an example of a structured set; here we list only a few representative ones (and perhaps also some exceptions).</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sets">sets</a> themselves (trivially);</li> <li><a class="existingWikiWord" href="/nlab/show/groups">groups</a>, <a class="existingWikiWord" href="/nlab/show/rings">rings</a>, and other objects of <a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a>;</li> <li><a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> and other concrete (based on sets of <a class="existingWikiWord" href="/nlab/show/points">points</a>) notions of <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a>;</li> <li>some other notions of <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> such as <a class="existingWikiWord" href="/nlab/show/locales">locales</a>, as long as we only consider isomorphisms, but <em>not</em> using functions as the other morphisms;</li> <li><em>not</em> some other notions of spaces such as <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> (at least if defined via topological spaces);</li> <li><a class="existingWikiWord" href="/nlab/show/strict+categories">strict categories</a> (where the underlying set is the set of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>) but <em>not</em> <a class="existingWikiWord" href="/nlab/show/categories">categories</a> treated up to <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a>.</li> </ul> <h2 id="structured_objects">Structured objects</h2> <p>Given any category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> whatsoever, we may define a <strong>type of structure</strong> on objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> as a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><msub><mi>S</mi> <mo>≅</mo></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">T\colon S_\cong \to Set</annotation></semantics></math> to <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> from the underyling groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. Then any <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful 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>C</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">U\colon C \to S</annotation></semantics></math> whatsoever defines a type of structure on objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> (at least if its fibres are essentially small), in the same way as a concrete category defines a type of structure on sets. Indeed, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> presents the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> as objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> with <strong><a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a></strong>.</p> <h2 id="see_also">See also</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+of+elements">category of elements</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 10, 2023 at 14:37:13. 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