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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>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="linear_algebra">Linear algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#rules'>Rules</a></li> <li><a href='#in_dagger_categories'>In dagger categories</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>In a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/biproducts">biproducts</a>, <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> between <a class="existingWikiWord" href="/nlab/show/finite+product">finite</a> biproducts are naturally encoded in terms of arrays of morphisms between the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct summands</a> of the <a class="existingWikiWord" href="/nlab/show/objects">objects</a>. The natural operations on morphisms (addition, composition) correspond to the usual matrix calculus operations on these arrays.</p> <p>For the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> this reproduces the standard matrix calculus of linear algebra.</p> <h2 id="rules">Rules</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in a <a class="existingWikiWord" href="/nlab/show/category">category</a> with <a class="existingWikiWord" href="/nlab/show/biproducts">biproducts</a> where the objects <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> are given as <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msubsup><mo>⊕</mo> <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow> <mi>m</mi></msubsup><msub><mi>X</mi> <mi>j</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>Y</mi><mo>=</mo><msubsup><mo>⊕</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><msub><mi>Y</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X = \oplus_{j = 1}^m X_j \,, \;\; Y = \oplus_{i = 1}^n Y_i \,. </annotation></semantics></math></div> <p>Since a <a class="existingWikiWord" href="/nlab/show/biproduct">biproduct</a> is both a <a class="existingWikiWord" href="/nlab/show/product">product</a> as well as a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is fixed by all its compositions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mi>j</mi> <mi>i</mi></msubsup></mrow><annotation encoding="application/x-tex">f^i_j</annotation></semantics></math> with the product projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>π</mi> <mi>i</mi></msup><mo>:</mo><mi>Y</mi><mo>→</mo><msub><mi>Y</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\pi^i : Y \to Y_i</annotation></semantics></math> and the coproduct injections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>j</mi></msub><mo>:</mo><msub><mi>X</mi> <mi>j</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\iota_j : X_j \to X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mi>j</mi> <mi>i</mi></msubsup><mo>:</mo><mo>=</mo><msub><mi>X</mi> <mi>j</mi></msub><mover><mo>→</mo><mrow><msub><mi>ι</mi> <mi>j</mi></msub></mrow></mover><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mover><mo>→</mo><mrow><msup><mi>π</mi> <mi>i</mi></msup></mrow></mover><msub><mi>Y</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^i_j := X_j \stackrel{\iota_j}{\to} X \stackrel{f}{\to} Y \stackrel{\pi^i}{\to} Y_i \,. </annotation></semantics></math></div> <p>In <strong>matrix calculus</strong> one therefore writes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><msubsup><mi>f</mi> <mn>1</mn> <mn>1</mn></msubsup></mtd> <mtd><msubsup><mi>f</mi> <mn>2</mn> <mn>1</mn></msubsup></mtd> <mtd><mi>⋯</mi></mtd> <mtd><msubsup><mi>f</mi> <mi>m</mi> <mn>1</mn></msubsup></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd> <mtd><mi>⋮</mi></mtd> <mtd><mi>⋱</mi></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><msubsup><mi>f</mi> <mn>1</mn> <mi>n</mi></msubsup></mtd> <mtd><msubsup><mi>f</mi> <mn>2</mn> <mi>n</mi></msubsup></mtd> <mtd><mi>⋯</mi></mtd> <mtd><msubsup><mi>f</mi> <mi>m</mi> <mi>n</mi></msubsup></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f := \left( \array{ f^1_1 & f^1_2 & \cdots & f^1_m \\ \vdots & \vdots & \ddots & \vdots \\ f^n_1 & f^n_2 & \cdots & f^n_m } \right) \,. </annotation></semantics></math></div> <p>With this notation one has the following rules for computation:</p> <ul> <li> <p>matrix addition, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g : X \to Y</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>+</mo><mi>g</mi><msubsup><mo stretchy="false">)</mo> <mi>j</mi> <mi>i</mi></msubsup><mo>=</mo><msubsup><mi>f</mi> <mi>j</mi> <mi>i</mi></msubsup><mo>+</mo><msubsup><mi>g</mi> <mi>j</mi> <mi>i</mi></msubsup></mrow><annotation encoding="application/x-tex"> (f + g)^i_j = f^i_j + g^i_j </annotation></semantics></math></div></li> <li> <p>matrix multiplication, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f:X \to Y</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">g:Y \to Z</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><msubsup><mo stretchy="false">)</mo> <mi>j</mi> <mi>i</mi></msubsup><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>k</mi></munder><msubsup><mi>g</mi> <mi>k</mi> <mi>i</mi></msubsup><mo>∘</mo><msubsup><mi>f</mi> <mi>j</mi> <mi>k</mi></msubsup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (g \circ f)^i_j = \sum_k g^i_k \circ f^k_j \,, </annotation></semantics></math></div></li> </ul> <p>where in each case the sum of morphisms is taken using the canonical <a class="existingWikiWord" href="/nlab/show/enriched+category">enrichment</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 abelian <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a> (as described at <a class="existingWikiWord" href="/nlab/show/biproduct">biproduct</a>).</p> <p>As can be seen in the above formulas, particularly for matrix multiplication, this is a context is which the <a class="existingWikiWord" href="/nlab/show/Einstein+summation+convention">Einstein summation convention</a> can be used, with a distinction drawn between upper and lower indices. Then repeated indices (in formulas with general applicability) will always appear once upper and once lower, summed over. However, this convention can apply only to the morphisms, not to the objects.</p> <h2 id="in_dagger_categories">In dagger categories</h2> <p>If 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> is in addition a <a class="existingWikiWord" href="/nlab/show/dagger+category">dagger category</a> with an obvious compatibility condition between the dagger operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>†</mo></msup><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">(-)^\dagger : C \to C</annotation></semantics></math> and the biproduct structure, then the usual rules of computation for matrices over complex numbers have analogs in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <ul> <li> <p>conjugation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>†</mo></msup><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mrow><mi>j</mi><mi>i</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mo>†</mo></msup></mrow><annotation encoding="application/x-tex"> (f^\dagger)_{i j} = (f_{j i})^\dagger </annotation></semantics></math></div></li> </ul> <p>Here the distinction between upper and lower indices cannot be maintained, although it is still true that repeated indices will be summed in formulas with general applicability.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+analysis">matrix analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+transform">integral transform</a></p> </li> </ul> <h2 id="references">References</h2> <p>Historical origins:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Arthur+Cayley">Arthur Cayley</a>: <em>A Memoir on the Theory of Matrices</em>, Philosophical Transactions of the Royal Society of London, <strong>148</strong> (1858) 17-37 [<a href="https://www.jstor.org/stable/108649">jstor:108649</a>]</li> </ul> <p>Discussion in the generality of <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Harding">John Harding</a>, section 2 of <em><a class="existingWikiWord" href="/nlab/files/matrixcalculus.pdf" title="Orthomodularity in dagger biproduct categories">Orthomodularity in dagger biproduct categories</a></em></li> </ul> <p>Formalization in terms of <a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">dependent linear type theory</a> is in</p> <ul> <li><em><a class="existingWikiWord" href="/schreiber/show/Type-semantics+for+quantization">Type-semantics for quantization</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 9, 2024 at 10:57:13. 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