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<p><strong><a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a> (<a class="existingWikiWord" href="/nlab/show/definitional+equality">definitional</a>, <a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional</a>, <a class="existingWikiWord" href="/nlab/show/computational+equality">computational</a>, <a class="existingWikiWord" href="/nlab/show/judgemental+equality">judgemental</a>, <a class="existingWikiWord" href="/nlab/show/extensional+equality">extensional</a>, <a class="existingWikiWord" href="/nlab/show/intensional+equality">intensional</a>, <a class="existingWikiWord" href="/nlab/show/decidable+equality">decidable</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a>, <a class="existingWikiWord" href="/nlab/show/definitional+isomorphism">definitional isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a>, <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+equivalence">gauge equivalence</a></p> </li> <li> <p><strong>Examples.</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>, <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a>, <a class="existingWikiWord" href="/nlab/show/weak+equivalence+of+internal+categories">weak equivalence of internal categories</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/principle+of+equivalence">principle of equivalence</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/equation">equation</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><strong>Examples.</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+equation">linear equation</a>, <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a>, <a class="existingWikiWord" href="/nlab/show/ordinary+differential+equation">ordinary differential equation</a>, <a class="existingWikiWord" href="/nlab/show/critical+locus">critical locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/Einstein+equation">Einstein equation</a>, <a class="existingWikiWord" href="/nlab/show/wave+equation">wave equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+equation">Schrödinger equation</a>, <a class="existingWikiWord" href="/nlab/show/Knizhnik-Zamolodchikov+equation">Knizhnik-Zamolodchikov equation</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+equation">Maurer-Cartan equation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Arnold+equation">Euler-Arnold equation</a>, <a class="existingWikiWord" href="/nlab/show/Fuchsian+equation">Fuchsian equation</a>, <a class="existingWikiWord" href="/nlab/show/Fokker-Planck+equation">Fokker-Planck equation</a>, <a class="existingWikiWord" href="/nlab/show/Lax+equation">Lax equation</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/equality+and+equivalence+-+contents">Edit this sidebar</a> </p> </div></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="equivalence">Equivalence</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#in_higher_category_theory'>In higher category theory</a></li> <li><a href='#in_homotopy_theory'>In homotopy theory</a></li> <li><a href='#in_homotopy_type_theory'>In homotopy type theory</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In general two objects are considered <em>equivalent</em> if they may be replaced by one another in all contexts under consideration (see also the <em><a class="existingWikiWord" href="/nlab/show/principle+of+equivalence">principle of equivalence</a></em>).</p> <p>At the propositional level, the general theory of equivalence is discussed at <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a>, but that leaves us with the question of the correct definition of equivalence in various situations. Furthermore, we can work on a higher level and ask what the equivalences are, not just whether things are equivalent.</p> <h2 id="in_higher_category_theory">In higher category theory</h2> <p>In <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> an <strong>equivalence</strong> often means a morphism which is invertible in a maximally weak sense (that is, up to all higher equivalences). Two objects are <strong>equivalent</strong> if there is an equivalence between them. For example:</p> <ul> <li> <p>In a <a class="existingWikiWord" href="/nlab/show/set">set</a> (a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/0-category">category</a>), equivalence is just <a class="existingWikiWord" href="/nlab/show/equality">equality</a>.</p> </li> <li> <p>In a <a class="existingWikiWord" href="/nlab/show/category">category</a> (a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/1-category">category</a>), an equivalence is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </li> <li> <p>In a (possibly weak) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-category">category</a>, an equivalence is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : x \to y</annotation></semantics></math> such that there exists a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>y</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">g : y\to x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-cell isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>g</mi><mo>≅</mo><msub><mn>1</mn> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">f g \cong 1_y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>f</mi><mo>≅</mo><msub><mn>1</mn> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">g f \cong 1_x</annotation></semantics></math>. In the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> this reduces to the usual notion of <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> (although one must define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Cat</mo></mrow><annotation encoding="application/x-tex">\Cat</annotation></semantics></math> carefully using <a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a>s if one wants to avoid the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>).</p> </li> <li> <p>In a (possibly weak) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-category">category</a>, an equivalence is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : x \to y</annotation></semantics></math> such that there exists a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>y</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">g : y\to x</annotation></semantics></math> and 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>g</mi><mo>→</mo><msub><mn>1</mn> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">f g \to 1_y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>f</mi><mo>→</mo><msub><mn>1</mn> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">g f \to 1_x</annotation></semantics></math> which are equivalences in the appropriate hom-2-categories. In the (weak) 3-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Bicat</mi></mrow><annotation encoding="application/x-tex">Bicat</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/bicategory">bicategories</a>, pseudofunctors, pseudonatural transformations, and modifications, such equivalences are often called <em>biequivalences</em>, and that term is sometimes also used for an equivalence in any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-category.</p> </li> <li> <p>In a (possibly weak) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/infinity-category">category</a>, an equivalence is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : x \to y</annotation></semantics></math> such that there exists a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>y</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">g : y\to x</annotation></semantics></math> and 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>g</mi><mo>→</mo><msub><mn>1</mn> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">f g \to 1_y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>f</mi><mo>→</mo><msub><mn>1</mn> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">g f \to 1_x</annotation></semantics></math> which are equivalences in the appropriate hom-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories. This can be made rigorous as a <a class="existingWikiWord" href="/nlab/show/corecursion">corecursive definition</a>.</p> </li> </ul> <p>There is also the more structured/coherent notion of an <a class="existingWikiWord" href="/nlab/show/adjoint+equivalence">adjoint equivalence</a>, but one hopes to have a theorem that any equivalence can be improved to an adjoint one. This is known in some cases, including 2-categories, 3-categories, strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories, and some models for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories.</p> <p>In addition to this, there is the notion of <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> (and higher categories), which is on a separate page. But it is related: properly done, two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories are equivalent if and only if they're equivalent as objects in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories.</p> <h2 id="in_homotopy_theory">In homotopy theory</h2> <p>In the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of topological spaces, notions weaker than isomorphism become very important: <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> and <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>. The latter notion is generalized to <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> for objects in any <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>. There is also a notion of equivalence between model categories: <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>.</p> <p>These can be understood to some extent using higher categories. For example, topological spaces should be weakly homotopy-equivalent if and only if they have equivalent <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoid">fundamental infinity-groupoids</a>. Similarly, Quillen equivalent model categories give rise to equivalent <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-categories</a>.</p> <p>See <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalence in an (infinity,1)-category</a>.</p> <h2 id="in_homotopy_type_theory">In homotopy type theory</h2> <p>In <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> equivalences can be axiomatized as those <a class="existingWikiWord" href="/nlab/show/terms">terms</a> of <a class="existingWikiWord" href="/nlab/show/function+types">function types</a> all whose <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> are <a class="existingWikiWord" href="/nlab/show/contractible">contractible</a>.</p> <p>See <em><a class="existingWikiWord" href="/nlab/show/equivalence+in+homotopy+type+theory">equivalence in homotopy type theory</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principle+of+equivalence">principle of equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/identity+of+indiscernibles">identity of indiscernibles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> </li> <li> <p><strong>equivalence</strong>, <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+2-category">equivalence in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+%282%2C1%29-categories">equivalence of (2,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+equivalence">Bousfield equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+homotopy+type+theory">equivalence in homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> </li> </ul> <div> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math>symbol<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math>in <a class="existingWikiWord" href="/nlab/show/logic">logic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/element">element</a> <a class="existingWikiWord" href="/nlab/show/relation">relation</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mo>:</mo></mrow><annotation encoding="application/x-tex">\,:</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/type">typing</a> <a class="existingWikiWord" href="/nlab/show/relation">relation</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/equality">equality</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/entailment">entailment</a> / <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo></mrow><annotation encoding="application/x-tex">\top</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/true">true</a> / <a class="existingWikiWord" href="/nlab/show/top">top</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/false">false</a> / <a class="existingWikiWord" href="/nlab/show/bottom">bottom</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/implication">implication</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Leftrightarrow</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo></mrow><annotation encoding="application/x-tex">\not</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/negation">negation</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\neq</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/negation">negation</a> of <a class="existingWikiWord" href="/nlab/show/equality">equality</a> / <a class="existingWikiWord" href="/nlab/show/apartness">apartness</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∉</mo></mrow><annotation encoding="application/x-tex">\notin</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/negation">negation</a> of <a class="existingWikiWord" href="/nlab/show/element">element</a> <a class="existingWikiWord" href="/nlab/show/relation">relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo></mrow><annotation encoding="application/x-tex">\not \not</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/double+negation">negation of negation</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∃</mo></mrow><annotation encoding="application/x-tex">\exists</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo></mrow><annotation encoding="application/x-tex">\forall</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\wedge</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\vee</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/logical+disjunction">logical disjunction</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong>symbol</strong></td><td style="text-align: left;"><strong>in <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</strong></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> (<a class="existingWikiWord" href="/nlab/show/implication">implication</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/product+type">product type</a> (<a class="existingWikiWord" href="/nlab/show/conjunction">conjunction</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow><annotation encoding="application/x-tex">+</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a> (<a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a> (<a class="existingWikiWord" href="/nlab/show/false">false</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a> (<a class="existingWikiWord" href="/nlab/show/true">true</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a> (<a class="existingWikiWord" href="/nlab/show/equality">equality</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a> (<a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo></mrow><annotation encoding="application/x-tex">\sum</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a> (<a class="existingWikiWord" href="/nlab/show/existential+quantifier">existential quantifier</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo></mrow><annotation encoding="application/x-tex">\prod</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> (<a class="existingWikiWord" href="/nlab/show/universal+quantifier">universal quantifier</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong>symbol</strong></td><td style="text-align: left;"><strong>in <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></strong></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊸</mo></mrow><annotation encoding="application/x-tex">\multimap</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/multiplicative+conjunction">multiplicative conjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/additive+disjunction">additive disjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>&</mi></mrow><annotation encoding="application/x-tex">\&</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/additive+conjunction">additive conjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⅋</mo></mrow><annotation encoding="application/x-tex">\invamp</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/multiplicative+disjunction">multiplicative disjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mo>!</mo></mrow><annotation encoding="application/x-tex">\;!</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/exponential+conjunction">exponential conjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mo>?</mo></mrow><annotation encoding="application/x-tex">\;?</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/exponential+disjunction">exponential disjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div></body></html> </div> <div class="revisedby"> <p> Last revised on October 3, 2021 at 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