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<title>Day convolution</title>
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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Day convolution</span></span>
</h1>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, specifically in <a href="Category_theory" title="Category theory">category theory</a>, <b>Day convolution</b> is an operation on <a href="Functor" title="Functor">functors</a> that can be seen as a <a href="Categorification" title="Categorification">categorified</a> version of <a href="Convolution" title="Convolution">function convolution</a>. It was first introduced by Brian Day in 1970<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> in the general context of <a href="Enriched_category" title="Enriched category">enriched</a> <a href="Functor_category" title="Functor category">functor categories</a>.
</p><p>Day convolution gives a <a href="Symmetric_monoidal_category" title="Symmetric monoidal category">symmetric monoidal</a> structure on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} (\mathbf {C} ,\mathbf {D} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi mathvariant="bold">D</mi>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} (\mathbf {C} ,\mathbf {D} )}</annotation>
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</math></span><img src="./4a7c93b88e615ab6132ffdf0a55f363a71f38700.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.666ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hom} (\mathbf {C} ,\mathbf {D} )}" loading="lazy"></span> for two symmetric monoidal categories <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} ,\mathbf {D} }">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
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<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">D</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} ,\mathbf {D} }</annotation>
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</math></span><img src="./3c87a6ae60b381ae724f559f632c08cb6b7589cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.015ex; height:2.509ex;" alt="{\displaystyle \mathbf {C} ,\mathbf {D} }" loading="lazy"></span>.
</p><p>Another related version is that Day convolution acts as a tensor product for a <a href="Monoidal_category" title="Monoidal category">monoidal category</a> structure on the category of functors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbf {C} ,V]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
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<mo>,</mo>
<mi>V</mi>
<mo stretchy="false">]</mo>
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<annotation encoding="application/x-tex">{\displaystyle [\mathbf {C} ,V]}</annotation>
</semantics>
</math></span><img src="./5cefe6959c0bae6a141dc8ba8444b96548c91fcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.046ex; height:2.843ex;" alt="{\displaystyle [\mathbf {C} ,V]}" loading="lazy"></span> over some <a href="Monoidal_category" title="Monoidal category">monoidal category</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<div class="mw-heading mw-heading3"><h3 id="First_version">First version</h3></div>
<p>Given <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F,G\colon \mathbf {C} \to \mathbf {D} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo>,</mo>
<mi>G</mi>
<mo>:<!-- : --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
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<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">D</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F,G\colon \mathbf {C} \to \mathbf {D} }</annotation>
</semantics>
</math></span><img src="./775a6456d6db3078fb38ee639875b525b69f42ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.23ex; height:2.509ex;" alt="{\displaystyle F,G\colon \mathbf {C} \to \mathbf {D} }" loading="lazy"></span> for two symmetric monoidal <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} ,\mathbf {D} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">D</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} ,\mathbf {D} }</annotation>
</semantics>
</math></span><img src="./3c87a6ae60b381ae724f559f632c08cb6b7589cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.015ex; height:2.509ex;" alt="{\displaystyle \mathbf {C} ,\mathbf {D} }" loading="lazy"></span>, we define their Day convolution as follows.
</p><p>It is the left <a href="Kan_extension" title="Kan extension">kan extension</a> along <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} \times \mathbf {C} \to ^{\otimes }\mathbf {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
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<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
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<mo>⊗<!-- ⊗ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} \times \mathbf {C} \to ^{\otimes }\mathbf {C} }</annotation>
</semantics>
</math></span><img src="./2a25c6b5a770d52e978e71bcbc0944b76a9fff12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.759ex; height:2.509ex;" alt="{\displaystyle \mathbf {C} \times \mathbf {C} \to ^{\otimes }\mathbf {C} }" loading="lazy"></span> of the composition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} \times \mathbf {C} \to ^{F,G}\mathbf {D} \times \mathbf {D} \to ^{\otimes }\mathbf {D} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
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<mi mathvariant="bold">C</mi>
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<mo stretchy="false">→<!-- → --></mo>
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<mi>F</mi>
<mo>,</mo>
<mi>G</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">D</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">D</mi>
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<msup>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊗<!-- ⊗ --></mo>
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</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">D</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} \times \mathbf {C} \to ^{F,G}\mathbf {D} \times \mathbf {D} \to ^{\otimes }\mathbf {D} }</annotation>
</semantics>
</math></span><img src="./a7c5d9efc10ec56837d824a06ecd2fd4b0bf398f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.644ex; height:2.676ex;" alt="{\displaystyle \mathbf {C} \times \mathbf {C} \to ^{F,G}\mathbf {D} \times \mathbf {D} \to ^{\otimes }\mathbf {D} }" loading="lazy"></span>
</p><p>Thus evaluated on an object <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\in \mathbf {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>O</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle O\in \mathbf {C} }</annotation>
</semantics>
</math></span><img src="./ed8bce380318703c67d998ad59ba6f42f2ba24d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.545ex; height:2.176ex;" alt="{\displaystyle O\in \mathbf {C} }" loading="lazy"></span>, intuitively we get a <a href="Colimit" class="mw-redirect" title="Colimit">colimit</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {D} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">D</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {D} }</annotation>
</semantics>
</math></span><img src="./b2345293072878db24e119c580def49ad582e3ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.05ex; height:2.176ex;" alt="{\displaystyle \mathbf {D} }" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)\otimes G(y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>⊗<!-- ⊗ --></mo>
<mi>G</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(x)\otimes G(y)}</annotation>
</semantics>
</math></span><img src="./24bd3f48b351b5253fe50bcd98e8bfb3aecdee9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.512ex; height:2.843ex;" alt="{\displaystyle F(x)\otimes G(y)}" loading="lazy"></span> along approximations of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O\in \mathbf {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>O</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle O\in \mathbf {C} }</annotation>
</semantics>
</math></span><img src="./ed8bce380318703c67d998ad59ba6f42f2ba24d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.545ex; height:2.176ex;" alt="{\displaystyle O\in \mathbf {C} }" loading="lazy"></span> as a pure tensor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\otimes y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>⊗<!-- ⊗ --></mo>
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\otimes y}</annotation>
</semantics>
</math></span><img src="./56efedc83f0f00d1a89451a753d743db31ceb74c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x\otimes y}" loading="lazy"></span>
</p><p>Left kan extensions are computed via <a href="Coend" class="mw-redirect" title="Coend">coends</a>, which leads to the version below.
</p>
<div class="mw-heading mw-heading3"><h3 id="Enriched_version">Enriched version</h3></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {C} ,\otimes _{c})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo>,</mo>
<msub>
<mo>⊗<!-- ⊗ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>c</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\mathbf {C} ,\otimes _{c})}</annotation>
</semantics>
</math></span><img src="./adfa995131d573cf8a58bd360e2ab51cdce95102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.527ex; height:2.843ex;" alt="{\displaystyle (\mathbf {C} ,\otimes _{c})}" loading="lazy"></span> be a monoidal category enriched over a symmetric <a href="Monoidal_closed_category" class="mw-redirect" title="Monoidal closed category">monoidal closed category</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V,\otimes )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>V</mi>
<mo>,</mo>
<mo>⊗<!-- ⊗ --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (V,\otimes )}</annotation>
</semantics>
</math></span><img src="./cb75fea2f6d767aa177776826f6b7327edc81cad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.439ex; height:2.843ex;" alt="{\displaystyle (V,\otimes )}" loading="lazy"></span>. Given two functors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F,G\colon \mathbf {C} \to V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo>,</mo>
<mi>G</mi>
<mo>:<!-- : --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F,G\colon \mathbf {C} \to V}</annotation>
</semantics>
</math></span><img src="./3ca50180b90496be956dab549056b67db7897297.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.968ex; height:2.509ex;" alt="{\displaystyle F,G\colon \mathbf {C} \to V}" loading="lazy"></span>, we define their Day convolution as the following coend.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\otimes _{d}G=\int ^{x,y\in \mathbf {C} }\mathbf {C} (x\otimes _{c}y,-)\otimes Fx\otimes Gy}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<msub>
<mo>⊗<!-- ⊗ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
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</msub>
<mi>G</mi>
<mo>=</mo>
<msup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mo>,</mo>
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<mo>∈<!-- ∈ --></mo>
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<mi mathvariant="bold">C</mi>
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</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<msub>
<mo>⊗<!-- ⊗ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>c</mi>
</mrow>
</msub>
<mi>y</mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mo stretchy="false">)</mo>
<mo>⊗<!-- ⊗ --></mo>
<mi>F</mi>
<mi>x</mi>
<mo>⊗<!-- ⊗ --></mo>
<mi>G</mi>
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F\otimes _{d}G=\int ^{x,y\in \mathbf {C} }\mathbf {C} (x\otimes _{c}y,-)\otimes Fx\otimes Gy}</annotation>
</semantics>
</math></span><img src="./88654f19e681d60ebb96b864e671698a5ce1e32a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.944ex; height:6.343ex;" alt="{\displaystyle F\otimes _{d}G=\int ^{x,y\in \mathbf {C} }\mathbf {C} (x\otimes _{c}y,-)\otimes Fx\otimes Gy}" loading="lazy"></span></dd></dl>
<p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes _{c}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>⊗<!-- ⊗ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>c</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \otimes _{c}}</annotation>
</semantics>
</math></span><img src="./b1c8a1ccde67f5c18a26e233f0e73c7ab0d22142.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.752ex; height:2.343ex;" alt="{\displaystyle \otimes _{c}}" loading="lazy"></span> is symmetric, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes _{d}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>⊗<!-- ⊗ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \otimes _{d}}</annotation>
</semantics>
</math></span><img src="./c589196d8de1a020f970408cb24f013235e70aeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.9ex; height:2.343ex;" alt="{\displaystyle \otimes _{d}}" loading="lazy"></span> is also symmetric. We can show this defines an associative monoidal product:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&amp;(F\otimes _{d}G)\otimes _{d}H\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2}}(F\otimes _{d}G)c_{1}\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2}}\left(\int ^{c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\right)\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{2}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{3}}Fc_{3}\otimes (G\otimes _{d}H)c_{1}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&amp;F\otimes _{d}(G\otimes _{d}H)\end{aligned}}}">
<semantics>
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<mtd></mtd>
<mtd>
<mi></mi>
<mo stretchy="false">(</mo>
<mi>F</mi>
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<mn>3</mn>
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<mn>4</mn>
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<mi>d</mi>
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<mi>G</mi>
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<mi>d</mi>
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<mi>H</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&amp;(F\otimes _{d}G)\otimes _{d}H\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2}}(F\otimes _{d}G)c_{1}\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2}}\left(\int ^{c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\right)\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{2}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{3}}Fc_{3}\otimes (G\otimes _{d}H)c_{1}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&amp;F\otimes _{d}(G\otimes _{d}H)\end{aligned}}}</annotation>
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</math></span><img src="./5539d6a3ddbe543a5a1691cc60953d17a5765282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -24.505ex; width:72.677ex; height:50.176ex;" alt="{\displaystyle {\begin{aligned}&amp;(F\otimes _{d}G)\otimes _{d}H\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2}}(F\otimes _{d}G)c_{1}\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2}}\left(\int ^{c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\right)\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4}\otimes _{c}c_{2},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{2}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&amp;\int ^{c_{1},c_{3}}Fc_{3}\otimes (G\otimes _{d}H)c_{1}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&amp;F\otimes _{d}(G\otimes _{d}H)\end{aligned}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFDay1970" class="citation journal cs1">Day, Brian (1970). "On closed categories of functors". <i>Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics</i>. <b>139</b>: <span class="nowrap">1–</span>38.</cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFLoregian2021" class="citation book cs1">Loregian, Fosco (2021). <i>(Co)end Calculus</i>. p.&nbsp;51. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1501.02503">1501.02503</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2F9781108778657">10.1017/9781108778657</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>9781108778657</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&nbsp;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:237839003">237839003</a>.</cite></span>
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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/Day+convolution">Day convolution</a> at the <a href="NLab" title="NLab"><i>n</i>Lab</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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