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<p><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></p> </li> </ul> <p><strong>With duals for objects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a></p> </li> </ul> <p><strong>With duals for morphisms</strong></p> <ul> <li> <p><span class="newWikiWord">monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a></p> </li> </ul> <p><strong>With traces</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a></p> </li> </ul> <p><strong>Closed structure</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a></p> </li> </ul> <p><strong>Special sorts of products</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> </li> </ul> <p><strong>Internal monoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> </ul> <p><strong id="_examples">Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <p><strong>In higher category theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</a></p> </li> </ul> </div></div> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <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> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+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='#examples'>Examples</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#SmashMonoidalDiagonals'>Smash-monoidal diagonals</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A general <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,\otimes)</annotation></semantics></math> does not admit <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⟶</mo><mi>x</mi><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x \longrightarrow x\otimes x</annotation></semantics></math>, unlike the case of a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> (where the monoidal product is the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a>). A <strong>monoidal category with diagonals</strong> is a monoidal category with the extra <a class="existingWikiWord" href="/nlab/show/structure">structure</a> of a consistent system of such <a class="existingWikiWord" href="/nlab/show/diagonal+morphisms">diagonal morphisms</a>.</p> <h2 id="definition">Definition</h2> <p>A consistent system of diagonal maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>x</mi></msub><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>→</mo><mi>x</mi><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta_x\colon x \to x \otimes x</annotation></semantics></math> as <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> varies through the <a class="existingWikiWord" href="/nlab/show/objects">objects</a> of a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,\otimes,I)</annotation></semantics></math> should be <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>⊗</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>Δ</mi> <mi>x</mi></msub><mo>=</mo><msub><mi>Δ</mi> <mi>y</mi></msub><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">(f \otimes f)\circ \Delta_x = \Delta_y \circ f</annotation></semantics></math>, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f\colon x\to y</annotation></semantics></math>. Hence such a system is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> from the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> on <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> to the <a class="existingWikiWord" href="/nlab/show/composition">composite</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>C</mi><mo>×</mo><mi>C</mi><mover><mo>→</mo><mo>⊗</mo></mover><mi>C</mi></mrow><annotation encoding="application/x-tex">C \to C \times C \stackrel{\otimes}{\to} C</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/diagonal+functor">diagonal functor</a> with the given <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal product functor</a>.</p> <p>Another desirable property is that the diagonal map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>I</mi></msub><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>I</mi><mo>⊗</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\Delta_I\colon I \to I\otimes I</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> <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> is the <a class="existingWikiWord" href="/nlab/show/inverse+morphism">inverse</a> of the left <a class="existingWikiWord" href="/nlab/show/unitor">unitor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mi>I</mi></msub><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>⊗</mo><mi>I</mi><mover><mo>→</mo><mo>∼</mo></mover><mi>I</mi></mrow><annotation encoding="application/x-tex">\ell_I\colon I \otimes I \stackrel{\sim}{\to} I</annotation></semantics></math> (which is the same as the right unitor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">r_I</annotation></semantics></math>).</p> <h2 id="examples">Examples</h2> <h3 id="general">General</h3> <ul> <li> <p>Any <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> has a canonical structure of diagonal maps given by the actual <a class="existingWikiWord" href="/nlab/show/diagonal+morphisms">diagonal morphisms</a>.</p> </li> <li> <p>Given a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a>, the <a class="existingWikiWord" href="/nlab/show/wide+subcategory">wide subcategory</a> consisting of the <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> is a monoidal category with the same product and unit. The <a class="existingWikiWord" href="/nlab/show/diagonal+morphism">diagonal morphism</a> of the original category, always a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>, plays the role of the diagonal maps in the subcategory. Now there are no projections, since, for instance, in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, the projection maps are almost never injective.</p> </li> </ul> <div> <h3 id="SmashMonoidalDiagonals">Smash-monoidal diagonals</h3> <p>Write</p> <div class="maruku-equation" id="eq:SmashMonoidalCategoryOfPointedTopologicalSpaces"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>PointedTopologicalSpaces</mi><mo>,</mo><msup><mi>S</mi> <mn>0</mn></msup><mo>,</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>SymmetricMonoidalCategories</mi></mrow><annotation encoding="application/x-tex"> \big( PointedTopologicalSpaces, S^0, \wedge \big) \;\;\in\; SymmetricMonoidalCategories </annotation></semantics></math></div> <ul> <li> <p>for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> (with respect to some <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a> such as <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> or <a class="existingWikiWord" href="/nlab/show/D-topological+spaces">D-topological spaces</a>)</p> </li> <li> <p>regarded as a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with tensor product the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> and unit the <a class="existingWikiWord" href="/nlab/show/0-sphere">0-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mo>*</mo> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">S^0 \,=\, \ast_+</annotation></semantics></math>.</p> </li> </ul> <p>This category also has a <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a>, given on pointed spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>𝒳</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X_i = (\mathcal{X}_i, x_i)</annotation></semantics></math> with underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒳</mi> <mi>i</mi></msub><mo>∈</mo><mi>TopologicalSpaces</mi></mrow><annotation encoding="application/x-tex">\mathcal{X}_i \in TopologicalSpaces</annotation></semantics></math> by</p> <div class="maruku-equation" id="eq:CartesianProductOfPointedTopologicalSpaces"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>𝒳</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><msub><mi>𝒳</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>𝒳</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>𝒳</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X_1 \times X_2 \;=\; (\mathcal{X}_1, x_1) \times (\mathcal{X}_2, x_2) \;\coloneqq\; \big( \mathcal{X}_1 \times \mathcal{X}_2 , (x_1, x_2) \big) \,. </annotation></semantics></math></div> <p>But since this <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> is a non-trivial <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient</a> of the Cartesian product</p> <div class="maruku-equation" id="eq:SmashProductOfPointedSpacesQuotientDefinition"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>X</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mfrac><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex"> X_1 \wedge X_1 \,\coloneqq\, \frac{X_1 \times X_2}{ X_1 \vee X_2 } </annotation></semantics></math></div> <p>it is not itself <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian</a>, but just <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a>.</p> <p>However, via the quotienting <a class="maruku-eqref" href="#eq:SmashProductOfPointedSpacesQuotientDefinition">(3)</a>, it still inherits, from the <a class="existingWikiWord" href="/nlab/show/diagonal+morphisms">diagonal morphisms</a> on underlying topological spaces</p> <div class="maruku-equation" id="eq:CartesianDiagonalOnTopologicalSpaces"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒳</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>Δ</mi> <mi>𝒳</mi></msub></mrow></mover></mtd> <mtd><mi>𝒳</mi><mo>×</mo><mi>𝒳</mi></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{X} &\overset{ \Delta_{\mathcal{X}} }{\longrightarrow}& \mathcal{X} \times \mathcal{X} \\ x &\mapsto& (x,x) } </annotation></semantics></math></div> <p>a suitable notion of <a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal diagonals</a>:</p> <p> <div class="num_defn" id="SmashMonoidalDiagonal"> <h6>Definition</h6> <p>[Smash monoidal diagonals]</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>PointedTopologicalSpaces</mi></mrow><annotation encoding="application/x-tex">X \,\in\, PointedTopologicalSpaces</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>X</mi><mo>∧</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">D_X \;\colon\; X \longrightarrow X \wedge X</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <svg xmlns="http://www.w3.org/2000/svg" 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x="110.154" y="11.045"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#I_5WIA5YrxLIArZT8vyv0quNZI4=-glyph-1-2" x="116.741" y="11.045"></use> </g> <path fill="none" stroke-width="0.478" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -0.00059375 -0.001625 L 21.761125 -0.001625 " transform="matrix(1, 0, 0, -1, 102.567, 12.764)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#I_5WIA5YrxLIArZT8vyv0quNZI4=-glyph-1-2" x="103.038" y="19.875"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#I_5WIA5YrxLIArZT8vyv0quNZI4=-glyph-7-2" x="110.624" y="19.875"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#I_5WIA5YrxLIArZT8vyv0quNZI4=-glyph-1-2" x="116.27" y="19.875"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#I_5WIA5YrxLIArZT8vyv0quNZI4=-glyph-5-2" x="127.529" y="14.479"></use> <use xlink:href="#I_5WIA5YrxLIArZT8vyv0quNZI4=-glyph-5-3" x="134.115491" y="14.479"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#I_5WIA5YrxLIArZT8vyv0quNZI4=-glyph-0-1" x="138.474" y="15.752"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#I_5WIA5YrxLIArZT8vyv0quNZI4=-glyph-6-2" x="151.786" y="15.752"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#I_5WIA5YrxLIArZT8vyv0quNZI4=-glyph-0-1" x="162.413" y="15.752"></use> </g> </svg> <p>of the Cartesian <a class="existingWikiWord" href="/nlab/show/diagonal+morphism">diagonal morphism</a> <a class="maruku-eqref" href="#eq:CartesianProductOfPointedTopologicalSpaces">(2)</a> with the <a class="existingWikiWord" href="/nlab/show/coprojection">coprojection</a> onto the defining <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> <a class="maruku-eqref" href="#eq:SmashProductOfPointedSpacesQuotientDefinition">(3)</a>.</p> <p></p> </div> </p> <p>It is immediate that:</p> <p> <div class="num_prop" id="SmashMonoidalDiaginalsAreMonoidalDiagonals"> <h6>Proposition</h6> <p>The smash monoidal diagonal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> (Def. <a class="maruku-ref" href="#SmashMonoidalDiagonal"></a>) makes the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <a class="maruku-eqref" href="#eq:SmashMonoidalCategoryOfPointedTopologicalSpaces">(1)</a> of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> with <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> a <a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a>, in that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>D</mi> <mrow><msup><mi>S</mi> <mn>0</mn></msup></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msup><mi>S</mi> <mn>0</mn></msup><mo>∧</mo><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">S^0 \overset{\;\;D_{S^0}\;\;}{\longrightarrow} S^0 \wedge S^0</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </li> </ol> <p></p> </div> </p> <p>While elementary in itself, this has the following profound consequence:</p> <p> <div class="num_remark"> <h6>Remark</h6> <p>[Suspension spectra have diagonals]</p> <p>Since the <a class="existingWikiWord" href="/nlab/show/suspension+spectrum">suspension spectrum</a>-<a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>PointedTopologicalSpaces</mi><mo>⟶</mo><mi>HighlyStructuredSpectra</mi></mrow><annotation encoding="application/x-tex"> \Sigma^\infty \;\colon\; PointedTopologicalSpaces \longrightarrow HighlyStructuredSpectra </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a> from <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> <a class="maruku-eqref" href="#eq:SmashMonoidalCategoryOfPointedTopologicalSpaces">(1)</a> to any standard category of <a class="existingWikiWord" href="/nlab/show/highly+structured+spectra">highly structured spectra</a> (by <a href="Introduction+to+Stable+homotopy+theory+--+1-2#SmashProductOfFreeSpectra">this Prop.</a>) it follows that <em><a class="existingWikiWord" href="/nlab/show/suspension+spectra">suspension spectra</a> have <a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal diagonals</a></em>, in the form of <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a></p> <div class="maruku-equation" id="eq:SmashMonoidalDiagonalOnSuspensionSpectra"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>Σ</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma^\infty X \overset{ \;\; \Sigma^\infty(D_X) \;\; }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big) </annotation></semantics></math></div> <p>to their respective <a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a>, which hence makes them into <em><a class="existingWikiWord" href="/nlab/show/comonoid+objects">comonoid objects</a></em>, namely <em><a class="existingWikiWord" href="/nlab/show/coring+spectra">coring spectra</a></em>.</p> <p>For example, given a <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology+theory">Whitehead-generalized cohomology theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde E</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represented</a> by a <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>E</mi><mo>,</mo><msup><mn>1</mn> <mi>E</mi></msup><mo>,</mo><msup><mi>m</mi> <mi>E</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>SymmetricMonoids</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>Spectra</mi><mo stretchy="false">)</mo><mo>,</mo><mi>𝕊</mi><mo>,</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \big(E, 1^E, m^E \big) \;\; \in \; SymmetricMonoids \big( Ho(Spectra), \mathbb{S}, \wedge \big) </annotation></semantics></math></div> <p>the smash-monoidal diagonal structure <a class="maruku-eqref" href="#eq:SmashMonoidalDiagonalOnSuspensionSpectra">(5)</a> on suspension spectra serves to define the <a class="existingWikiWord" href="/nlab/show/cup+product">cup 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><mo>∪</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-)\cup (-)</annotation></semantics></math> in the corresponding <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory structure</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">[</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>c</mi> <mi>i</mi></msub></mrow></mover><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mi>E</mi><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow></mrow> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>∪</mo><mo stretchy="false">[</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">[</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mover><mo>⟶</mo><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow></mover><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mi>E</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mi>E</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>⟶</mo><mrow><msup><mi>m</mi> <mi>E</mi></msup></mrow></mover><msup><mi>Σ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mi>E</mi><mo maxsize="1.8em" minsize="1.8em">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow></mrow> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \big[ \Sigma^\infty X \overset{c_i}{\longrightarrow} \Sigma^{n_i} E \big] \,\in\, {\widetilde E}{}^{n_i}(X) \\ & \Rightarrow \;\; [c_1] \cup [c_2] \, \coloneqq \, \Big[ \Sigma^\infty X \overset{ \Sigma^\infty(D_X) }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big) \overset{ ( c_1 \wedge c_2 ) }{\longrightarrow} \big( \Sigma^{n_1} E \big) \wedge \big( \Sigma^{n_2} E \big) \overset{ m^E }{\longrightarrow} \Sigma^{n_1 + n_2}E \Big] \;\; \in \, {\widetilde E}{}^{n_1+n_2}(X) \,. \end{aligned} </annotation></semantics></math></div> <p></p> </div> </p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/relevance+monoidal+category">relevance monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></p> </li> </ul> <h2 id="references">References</h2> <p>The stronger notion of relevance monoidal category is discussed in</p> <ul> <li>K. Dosen and Z. Petric, <em>Relevant Categories and Partial Functions</em>, Publications de l’Institut Mathématique, Nouvelle Série, Vol. 82(96), pp. 17–23 (2007) (<a href="http://arxiv.org/abs/math/0504133">arXiv:0504133</a></li> </ul> <p>When a <a class="existingWikiWord" href="/nlab/show/premonoidal+category">premonoidal category</a> comes equipped with a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>x</mi></msub><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>→</mo><mi>x</mi><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta_x\colon x\to x\otimes x</annotation></semantics></math> for all <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>, such as in the <a class="existingWikiWord" href="/nlab/show/Kleisli+category">Kleisli category</a> for a <a class="existingWikiWord" href="/nlab/show/strong+monad">strong monad</a> on a cartesian category, or in any <a class="existingWikiWord" href="/nlab/show/Freyd+category">Freyd category</a>, then the <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> for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>⊗</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>Δ</mi> <mi>x</mi></msub><mo>=</mo><msub><mi>Δ</mi> <mi>y</mi></msub><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">(f\otimes f)\circ \Delta_x = \Delta_y \circ f</annotation></semantics></math> are called “copyable”.</p> <ul> <li>C. Führmann. <em>Varieties of effects</em>, Proc. Fossacs 2002. Doi:<a href="https://doi.org/10.1007/3-540-45931-6_11">10.1007/3-540-45931-6_11</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 30, 2021 at 04:56:11. 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