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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="enriched_category_theory">Enriched category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>, <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>, <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>, <a class="existingWikiWord" href="/nlab/show/double+category">double category</a>, <a class="existingWikiWord" href="/nlab/show/virtual+double+category">virtual double category</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+adjoint+functor">enriched adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+product+category">enriched product category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a></p> </li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>, <a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> </ul> <h2 id="extra_stuff_structure_property">Extra stuff, structure, property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/copowering">copowering</a> (<a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/powering">powering</a> (<a class="existingWikiWord" href="/nlab/show/cotensoring">cotensoring</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+category">monoidal enriched category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+enriched+category">cartesian closed enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+enriched+category">locally cartesian closed enriched category</a></p> </li> </ul> </li> </ul> <h3 id="homotopical_enrichment">Homotopical enrichment</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+homotopical+category">enriched homotopical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">model structure on homotopical presheaves</a></p> </li> </ul> </div></div> <h4 id="mapping_space">Mapping space</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>/<a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></strong></p> <h3 id="general_abstract">General abstract</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a>, <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a>, <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>, <a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a>, <a class="existingWikiWord" href="/nlab/show/derived+hom-space">derived hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a>, <a class="existingWikiWord" href="/nlab/show/derived+loop+space">derived loop space</a></p> </li> </ul> <h3 id="topology">Topology</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> (<a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topology+of+mapping+spaces">topology of mapping spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/evaluation+fibration+of+mapping+spaces">evaluation fibration of mapping spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/free+loop+space+of+a+classifying+space">free loop space of a classifying space</a></li> </ul> </li> </ul> <h3 id="simplicial_homotopy_theory">Simplicial homotopy theory</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+mapping+complex">simplicial mapping complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inertia+groupoid">inertia groupoid</a></p> </li> </ul> <h3 id="differential_topology">Differential topology</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+topology+of+mapping+spaces">differential topology of mapping spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/C-k+topology">C-k topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/tangent+spaces+of+mapping+spaces">tangent spaces of mapping spaces</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+loop+space">smooth loop space</a></p> </li> </ul> <h3 id="stable_homotopy_theory">Stable homotopy theory</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/mapping+spectrum">mapping spectrum</a></li> </ul> <h3 id="geometric_homotopy_theory">Geometric homotopy theory</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inertia+stack">inertia stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+loop+stack">free loop stack</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/mapping+space+-+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='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>An ordinary <a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> has for any ordered pair of <a class="existingWikiWord" href="/nlab/show/object">object</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(x,y)</annotation></semantics></math>—an object in the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> more generally an <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> over a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math>, there is – by definition – for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,y</annotation></semantics></math> an <em>object</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>obj</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}(x,y) \in obj V</annotation></semantics></math> that plays the role of the “collection of morphisms” from <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> to <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>.</p> <p>Accordingly, if here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math> is a category of “spaces” of sorts, then one also speaks of a <em>hom space</em>. For instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math> =<a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/linear+category">linear category</a>, then hom-spaces are vector spaces.</p> <p>Or if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math>=<a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a>, then hom-spaces are <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. If this last example is regarded in the context of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, then one may consider just the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of these topological spaces, which is equivalently modeled as a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> (<a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>) and thought of as an <a class="existingWikiWord" href="/nlab/show/infinity-groupoid">infinity-groupoid</a>. In all these cases people still often speak of “hom spaces”, but see at <em><a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a></em>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>The category Prop of <a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>s is enriched over itself. Hence for any two proposition <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,b</annotation></semantics></math>, there is a hom-proposition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Prop</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Prop(A,B)</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/implication">implication</a> proposition <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 \implies b</annotation></semantics></math>.</p> </li> <li> <p>The category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of <a class="existingWikiWord" href="/nlab/show/set">set</a>s is enriched over itself. Hence for any two sets <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,B</annotation></semantics></math>, there is a hom-set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Set(A,B)</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/function+set">function set</a> <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 \to B</annotation></semantics></math>.</p> </li> <li> <p>The category <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> of <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>s is enriched over itself. Hence for any two groupoids <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,B</annotation></semantics></math>, there is a <a class="existingWikiWord" href="/nlab/show/hom-groupoid">hom-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grpd</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Grpd(A,B)</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(A,B)</annotation></semantics></math>.</p> </li> <li> <p>The category <a class="existingWikiWord" href="/nlab/show/Pos">Pos</a> of <a class="existingWikiWord" href="/nlab/show/poset">poset</a>s is enriched over itself. Hence for any two posets <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,B</annotation></semantics></math>, there is a hom-poset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pos</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pos(A,B)</annotation></semantics></math>, the poset of <a class="existingWikiWord" href="/nlab/show/monotone">monotone</a>s.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+hom-functor">enriched hom-functor</a></p> </li> <li> <p><strong>hom-object</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a>, <a class="existingWikiWord" href="/nlab/show/hom-group">hom-group</a>, <a class="existingWikiWord" href="/nlab/show/hom-category">hom-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+hom-space">derived hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+hom-functor">enriched hom-functor</a>, <a class="existingWikiWord" href="/nlab/show/internal+hom-functor">internal hom-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+hom-functor">derived hom-functor</a></p> </li> </ul> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></th><th><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/homology">homology</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[S^n,-]</annotation></semantics></math></td><td style="text-align: left;"><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><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,A]</annotation></semantics></math></td><td style="text-align: left;"><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><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(-) \otimes A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/covariant+functor">covariant</a> <a class="existingWikiWord" href="/nlab/show/hom+functor">hom</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant</a> <a class="existingWikiWord" href="/nlab/show/hom+functor">hom</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/end">end</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/end">end</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coend">coend</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(S^n,-)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(-,A)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derived+tensor+product">derived tensor product</a> <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><mo stretchy="false">)</mo><msup><mo>⊗</mo> <mi>𝕃</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">(-) \otimes^{\mathbb{L}} A</annotation></semantics></math></td></tr> </tbody></table> </div></body></html> </div> <div class="revisedby"> <p> Last revised on February 5, 2024 at 07:11:09. 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