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this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry is about scales in <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>. For scales in <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> and <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, see <a class="existingWikiWord" href="/nlab/show/length+scale">length scale</a>.</p> </blockquote> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The idea of a scale comes from <a class="existingWikiWord" href="/nlab/show/Peter+Freyd">Peter Freyd</a>, in his attempt to give an algebraic (in the sense of universal algebra) description of the real <a class="existingWikiWord" href="/nlab/show/interval">interval</a>, which he believed to be more fundamental than the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> themselves in <a class="existingWikiWord" href="/nlab/show/analysis">analysis</a>, and of concepts from analysis such as <a class="existingWikiWord" href="/nlab/show/Lipschitz+continuity">Lipschitz continuity</a>, <a class="existingWikiWord" href="/nlab/show/limits">limits</a>, <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a> and <a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>. Freyd also discovered that scales are models of multiplicative-additive <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a> with a midpoint operation.</p> <h2 id="definition">Definition</h2> <p>A <strong>scale</strong> is a <a class="existingWikiWord" href="/nlab/show/minor+scale">minor scale</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> satisfying the scale identities:</p> <ul> <li> <p>for all <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 <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⊗</mo><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi>a</mi> <mo>∨</mo></msup><mo>⊗</mo><msup><mi>b</mi> <mo>∧</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo stretchy="false">(</mo><msup><mi>a</mi> <mo>∧</mo></msup><mo>⊗</mo><msup><mi>b</mi> <mo>∨</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \otimes b = (a^\vee \otimes b^\wedge) \vert (a^\wedge \otimes b^\vee)</annotation></semantics></math></p> </li> <li> <p>for all <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 <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⊕</mo><mi>b</mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi>a</mi> <mo>∧</mo></msup><mo>⊕</mo><msup><mi>b</mi> <mo>∨</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo stretchy="false">(</mo><msup><mi>a</mi> <mo>∨</mo></msup><mo>⊕</mo><msup><mi>b</mi> <mo>∧</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \oplus b = (a^\wedge \oplus b^\vee) \vert (a^\vee \oplus b^\wedge)</annotation></semantics></math></p> </li> </ul> <h2 id="properties">Properties</h2> <p>Every scale with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo><mo>=</mo><mo>⊤</mo></mrow><annotation encoding="application/x-tex">\bot = \top</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/trivial+object">trivial</a>.</p> <p>As a scale is a <a class="existingWikiWord" href="/nlab/show/closed+midpoint+algebra">closed midpoint algebra</a>, a scale has a <a class="existingWikiWord" href="/nlab/show/partial+order">partial order</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math>. For all <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 <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, <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 \leq b</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⊸</mo><mi>b</mi><mo>=</mo><mo>⊤</mo></mrow><annotation encoding="application/x-tex">a \multimap b = \top</annotation></semantics></math>.</p> <p>A subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">\mathcal{I}</annotation></semantics></math> of a scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is an <strong><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a></strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo><mo>∈</mo><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">\bot \in \mathcal{I}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∨</mo><mi>y</mi><mo>∈</mo><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">x \vee y \in \mathcal{I}</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">x \in \mathcal{I}</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><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">y \in \mathcal{I}</annotation></semantics></math>. An ideal is a <strong>zoom-invariant ideal</strong> if it is closed under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math>-zooming.</p> <p>A subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">\mathcal{I}</annotation></semantics></math> of a scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math>-face</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo><mo>∈</mo><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">\bot \in \mathcal{I}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo stretchy="false">|</mo><mi>y</mi><mo>∈</mo><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">x \vert y \in \mathcal{I}</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">x \in \mathcal{I}</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><mi>ℐ</mi></mrow><annotation encoding="application/x-tex">y \in \mathcal{I}</annotation></semantics></math>.</p> <p>Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math>-face is a zoom-invariant ideal.</p> <p>Given an element <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> in scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, a <strong>principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math>-face</strong> <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>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((a))</annotation></semantics></math> is the subset of all <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⊥</mo><mo stretchy="false">|</mo><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mi>b</mi><mo>≤</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">(\bot\vert)^n b \leq a</annotation></semantics></math> for all large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>.</p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/Jacobson+radical">Jacobson radical</a></strong> of a scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J(M)</annotation></semantics></math> of all <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>, <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><mo>⊥</mo><mo stretchy="false">|</mo><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mo>⊤</mo></mrow><annotation encoding="application/x-tex">x \leq (\bot\vert)^n \top</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is <strong>semi-simple</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J(M)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/trivial">trivial</a>.</p> <div class="standout"> <p>The proof of the Linear Representation Theorem in section 8 of <em>Algebraic Real Analysis</em> by Peter Freyd requires the use of <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a> through its implicit definition of the <a class="existingWikiWord" href="/nlab/show/quasiorder">quasiorder</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo><</mo></mrow><annotation encoding="application/x-tex">\lt</annotation></semantics></math> from the algebraically defined <a class="existingWikiWord" href="/nlab/show/partial+order">partial order</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math>. In particular, that every scale is a <a class="existingWikiWord" href="/nlab/show/%2A-autonomous+category">*-autonomous category</a> and thus a model for <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a> and that every equational axiom added to the theory of minor scales is either a consequence of the scale identity for scales or is inconsistent with the theory of minor scales are classical results, as certain lemmas used in the proofs have only been derived from the scale identities through the Linear Representation Theorem. The same is true of the definition of simple scales in section 10, of the algebraic construction of the standard interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> from simple scales in section 11, and various results involving absolute retracts in section 25. Since quasiorders can be constructed from partial orders in any set with a <a class="existingWikiWord" href="/nlab/show/tight+apartness+relation">tight apartness relation</a>, these results hold if the scales have a <a class="existingWikiWord" href="/nlab/show/tight+apartness+relation">tight apartness relation</a>, but it is unknown if these results still hold for general scales in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>.</p> </div> <h2 id="examples">Examples</h2> <p>The <a class="existingWikiWord" href="/nlab/show/unit+interval">unit interval</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo stretchy="false">|</mo><mi>b</mi><mo>≔</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">a \vert b \coloneqq \frac{a + b}{2}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊙</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\odot = \frac{1}{2}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mo>•</mo></msup><mo>=</mo><mn>1</mn><mo>−</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">a^\bullet = 1 - a</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\bot = 0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\top = 1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mo>∧</mo></msup><mo>=</mo><mi>max</mi><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a^\wedge = max(2a-1,0)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mo>∨</mo></msup><mo>=</mo><mi>min</mi><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a^\vee = min(2a,1)</annotation></semantics></math> is an example of a scale.</p> <p>The set of truth values in Girard’s <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a> is a scale.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/minor+scale">minor scale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+scale">Heyting scale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Freyd">Peter Freyd</a>, <em>Algebraic real analysis</em>, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (<a href="http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html">tac:20-10</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 17, 2025 at 17:37:49. See the <a href="/nlab/history/scale" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/scale" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/13035/#Item_4">Discuss</a><span class="backintime"><a href="/nlab/revision/scale/17" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/scale" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/scale" accesskey="S" class="navlink" id="history" rel="nofollow">History (17 revisions)</a> <a href="/nlab/show/scale/cite" style="color: black">Cite</a> <a href="/nlab/print/scale" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/scale" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>