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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='category_theory'>Category theory</h4> <div class='hide'> <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/end'>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%E2%80%93Ulmer+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> <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='#properties'>Properties</a></li><li><a href='#see_also'>See also</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>In <a class='existingWikiWord' href='/nlab/show/set+theory'>set theory</a>, the <a class='existingWikiWord' href='/nlab/show/support+of+a+set'>support of a set</a> turns the set into a <a class='existingWikiWord' href='/nlab/show/subsingleton'>subsingleton</a>. In <a class='existingWikiWord' href='/nlab/show/dependent+type+theory'>dependent type theory</a>, <a class='existingWikiWord' href='/nlab/show/bracket+type'>propositional truncation</a> or the <a class='existingWikiWord' href='/nlab/show/bracket+type'>bracket type</a> is an operation which takes a <a class='existingWikiWord' href='/nlab/show/type'>type</a> and turns it into a <a class='existingWikiWord' href='/nlab/show/mere+proposition'>h-propositions</a>, which are the type theoretic equivalent of subsingletons. By the <a class='existingWikiWord' href='/nlab/show/relation+between+type+theory+and+category+theory'>relation between type theory and category theory</a> and the relation between set theory and <a class='existingWikiWord' href='/nlab/show/sheaf+and+topos+theory'>topos theory</a>, it should be possible to do the same process and turn an <a class='existingWikiWord' href='/nlab/show/object'>object</a> of a <a class='existingWikiWord' href='/nlab/show/category'>category</a> into a <a class='existingWikiWord' href='/nlab/show/subterminal+object'>subterminal object</a>. These are the <strong>support objects</strong>, <strong>bracket objects</strong>, or <strong>propositional truncation objects</strong> in category theory.</p> <h2 id='definition'>Definition</h2> <p>The support object <math class='maruku-mathml' display='inline' id='mathml_b9881619b1d9c39c3ebd17436a7256bf83071ab2_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X]</annotation></semantics></math> of an object <math class='maruku-mathml' display='inline' id='mathml_b9881619b1d9c39c3ebd17436a7256bf83071ab2_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> in a <a class='existingWikiWord' href='/nlab/show/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_b9881619b1d9c39c3ebd17436a7256bf83071ab2_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> with a <a class='existingWikiWord' href='/nlab/show/terminal+object'>terminal object</a> <math class='maruku-mathml' display='inline' id='mathml_b9881619b1d9c39c3ebd17436a7256bf83071ab2_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/image'>image</a> of the unique morphism <math class='maruku-mathml' display='inline' id='mathml_b9881619b1d9c39c3ebd17436a7256bf83071ab2_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>X \to 1</annotation></semantics></math>.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b9881619b1d9c39c3ebd17436a7256bf83071ab2_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mo stretchy='false'>[</mo><mi>X</mi><mo stretchy='false'>]</mo><mo>↪</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>X \to [X] \hookrightarrow 1</annotation></semantics></math></div> <p>As a result, that all support objects exist is equivalent to the condition that every morphism into the terminal object has an <a class='existingWikiWord' href='/nlab/show/image'>image factorization</a>.</p> <h2 id='properties'>Properties</h2> <p>In any <a class='existingWikiWord' href='/nlab/show/well-pointed+topos'>well-pointed pretopos</a> <math class='maruku-mathml' display='inline' id='mathml_b9881619b1d9c39c3ebd17436a7256bf83071ab2_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{E}</annotation></semantics></math>, the support is the <a class='existingWikiWord' href='/nlab/show/coequalizer'>coequalizer</a> of the <a class='existingWikiWord' href='/nlab/show/cartesian+product'>product</a> projection morphisms <math class='maruku-mathml' display='inline' id='mathml_b9881619b1d9c39c3ebd17436a7256bf83071ab2_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\pi_1:X \times X \to X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b9881619b1d9c39c3ebd17436a7256bf83071ab2_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\pi_2:X \times X \to X</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b9881619b1d9c39c3ebd17436a7256bf83071ab2_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><munder><mo>⟶</mo><mrow><msub><mi>π</mi> <mn>2</mn></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow></mover></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width='0'><mi>p</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy='false'>[</mo><mi>X</mi><mo stretchy='false'>]</mo></mtd></mtr></mtable></mrow><mo>.</mo></mrow><annotation encoding='application/x-tex'>\array{ X \times X && \stackrel{\overset{\pi_1}{\longrightarrow}}{\underset{\pi_2}{\longrightarrow}} && X \\ & \searrow && \swarrow_{\mathrlap{p}} \\ && [X] }. </annotation></semantics></math></div> <h2 id='see_also'>See also</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/support'>support</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/subsingleton'>subsingleton</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/subterminal+object'>subterminal object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/complete+loop+graph'>complete loop graph</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/cartesian+product'>product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/coequalizer'>coequalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/bracket+type'>propositional truncation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/inhabited+object'>inhabited object</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/computational+trilogy'>computational trinitarianism</a></strong> = \linebreak <strong><a class='existingWikiWord' href='/nlab/show/propositions+as+types'>propositions as types</a></strong> +<strong><a class='existingWikiWord' href='/nlab/show/proofs+as+programs'>programs as proofs</a></strong> +<strong><a class='existingWikiWord' href='/nlab/show/relation+between+type+theory+and+category+theory'>relation type theory/category theory</a></strong></p> <table><thead><tr><th><a class='existingWikiWord' href='/nlab/show/logic'>logic</a></th><th><a class='existingWikiWord' href='/nlab/show/set+theory'>set theory</a> (<a class='existingWikiWord' href='/nlab/show/internal+logic+of+set+theory'>internal logic</a> of)</th><th><a class='existingWikiWord' href='/nlab/show/category+theory'>category theory</a></th><th><a class='existingWikiWord' href='/nlab/show/type+theory'>type theory</a></th></tr></thead><tbody><tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/proposition'>proposition</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/set'>set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/object'>object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/type'>type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/proposition'>predicate</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/family+of+sets'>family of sets</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/display+map'>display morphism</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/dependent+type'>dependent type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/proof'>proof</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/element'>element</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/generalized+element'>generalized element</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/term'>term</a>/<a class='existingWikiWord' href='/nlab/show/program'>program</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/cut+rule'>cut rule</a></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/composition'>composition</a> of <a class='existingWikiWord' href='/nlab/show/classifying+morphism'>classifying morphisms</a> / <a class='existingWikiWord' href='/nlab/show/pullback'>pullback</a> of <a class='existingWikiWord' href='/nlab/show/display+map'>display maps</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/substitution'>substitution</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/natural+deduction'>introduction rule</a> for <a class='existingWikiWord' href='/nlab/show/implication'>implication</a></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/unit+of+an+adjunction'>counit</a> for hom-tensor adjunction</td><td style='text-align: left;'>lambda</td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/term+elimination'>elimination rule</a> for <a class='existingWikiWord' href='/nlab/show/implication'>implication</a></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/unit'>unit</a> for hom-tensor adjunction</td><td style='text-align: left;'>application</td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/cut+rule'>cut elimination</a> for <a class='existingWikiWord' href='/nlab/show/implication'>implication</a></td><td style='text-align: left;'></td><td style='text-align: left;'>one of the <a class='existingWikiWord' href='/nlab/show/triangle+identities'>zigzag identities</a> for hom-tensor adjunction</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/beta-reduction'>beta reduction</a></td></tr> <tr><td style='text-align: left;'>identity elimination for <a class='existingWikiWord' href='/nlab/show/implication'>implication</a></td><td style='text-align: left;'></td><td style='text-align: left;'>the other <a class='existingWikiWord' href='/nlab/show/triangle+identities'>zigzag identity</a> for hom-tensor adjunction</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/eta-conversion'>eta conversion</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/true+proposition'>true</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/singleton'>singleton</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/terminal+object'>terminal object</a>/<a class='existingWikiWord' href='/nlab/show/truncated+object'>(-2)-truncated object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/contractible+type'>h-level 0</a>-<a class='existingWikiWord' href='/nlab/show/type'>type</a>/<a class='existingWikiWord' href='/nlab/show/unit+type'>unit type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/falsehood'>false</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/empty+set'>empty set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/initial+object'>initial object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/empty+type'>empty type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/proposition'>proposition</a>, <a class='existingWikiWord' href='/nlab/show/truth+value'>truth value</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/subsingleton'>subsingleton</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/subterminal+object'>subterminal object</a>/<a class='existingWikiWord' href='/nlab/show/truncated+object'>(-1)-truncated object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/mere+proposition'>h-proposition</a>, <a class='existingWikiWord' href='/nlab/show/mere+proposition'>mere proposition</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/logical+conjunction'>logical conjunction</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/cartesian+product'>cartesian product</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/cartesian+product'>product</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/product+type'>product type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/disjunction'>disjunction</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/disjoint+union'>disjoint union</a> (<a class='existingWikiWord' href='/nlab/show/support'>support</a> of)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/coproduct'>coproduct</a> (<a class='existingWikiWord' href='/nlab/show/truncation'>(-1)-truncation</a> of)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/sum+type'>sum type</a> (<a class='existingWikiWord' href='/nlab/show/bracket+type'>bracket type</a> of)</td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/implication'>implication</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/function+set'>function set</a> (into <a class='existingWikiWord' href='/nlab/show/subsingleton'>subsingleton</a>)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/internal+hom'>internal hom</a> (into <a class='existingWikiWord' href='/nlab/show/subterminal+object'>subterminal object</a>)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/function+type'>function type</a> (into <a class='existingWikiWord' href='/nlab/show/mere+proposition'>h-proposition</a>)</td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/negation'>negation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/function+set'>function set</a> into <a class='existingWikiWord' href='/nlab/show/empty+set'>empty set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/internal+hom'>internal hom</a> into <a class='existingWikiWord' href='/nlab/show/initial+object'>initial object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/function+type'>function type</a> into <a class='existingWikiWord' href='/nlab/show/empty+type'>empty type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/universal+quantifier'>universal quantification</a></td><td style='text-align: left;'>indexed <a class='existingWikiWord' href='/nlab/show/cartesian+product'>cartesian product</a> (of family of <a class='existingWikiWord' href='/nlab/show/subsingleton'>subsingletons</a>)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/dependent+product'>dependent product</a> (of family of <a class='existingWikiWord' href='/nlab/show/subterminal+object'>subterminal objects</a>)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/dependent+product+type'>dependent product type</a> (of family of <a class='existingWikiWord' href='/nlab/show/mere+proposition'>h-propositions</a>)</td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/existential+quantifier'>existential quantification</a></td><td style='text-align: left;'>indexed <a class='existingWikiWord' href='/nlab/show/disjoint+union'>disjoint union</a> (<a class='existingWikiWord' href='/nlab/show/support'>support</a> of)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/dependent+sum'>dependent sum</a> (<a class='existingWikiWord' href='/nlab/show/truncation'>(-1)-truncation</a> of)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/dependent+sum+type'>dependent sum type</a> (<a class='existingWikiWord' href='/nlab/show/bracket+type'>bracket type</a> of)</td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/logical+equivalence'>logical equivalence</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/bijection+set'>bijection set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/object+of+isomorphisms'>object of isomorphisms</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/equivalence+type'>equivalence type</a></td></tr> <tr><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/support+of+a+set'>support set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/support+object'>support object</a>/<a class='existingWikiWord' href='/nlab/show/truncation'>(-1)-truncation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/bracket+type'>propositional truncation</a>/<a class='existingWikiWord' href='/nlab/show/bracket+type'>bracket type</a></td></tr> <tr><td style='text-align: left;'></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/n-image'>n-image</a> of <a class='existingWikiWord' href='/nlab/show/morphism'>morphism</a> into <a class='existingWikiWord' href='/nlab/show/terminal+object'>terminal object</a>/<a class='existingWikiWord' href='/nlab/show/truncation'>n-truncation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/n-truncation+modality'>n-truncation modality</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/equality'>equality</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diagonal+function'>diagonal function</a>/<a class='existingWikiWord' href='/nlab/show/diagonal+subset'>diagonal subset</a>/<a class='existingWikiWord' href='/nlab/show/equality'>diagonal relation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/path+space+object'>path space object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/identity+type'>identity type</a>/<a class='existingWikiWord' href='/nlab/show/path+type'>path type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/presentation+axiom'>completely presented set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/set'>set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/discrete+object'>discrete object</a>/<a class='existingWikiWord' href='/nlab/show/truncated+object'>0-truncated object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/h-set'>h-level 2</a>-<a class='existingWikiWord' href='/nlab/show/type'>type</a>/<a class='existingWikiWord' href='/nlab/show/set'>set</a>/<a class='existingWikiWord' href='/nlab/show/h-set'>h-set</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/set'>set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/set'>set</a> with <a class='existingWikiWord' href='/nlab/show/equivalence+relation'>equivalence relation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category'>internal 0-groupoid</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/Bishop+set'>Bishop set</a>/<a class='existingWikiWord' href='/nlab/show/setoid'>setoid</a> with its <a class='existingWikiWord' href='/nlab/show/pseudo-equivalence+relation'>pseudo-equivalence relation</a> an actual <a class='existingWikiWord' href='/nlab/show/equivalence+relation'>equivalence relation</a></td></tr> <tr><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/equivalence+class'>equivalence class</a>/<a class='existingWikiWord' href='/nlab/show/quotient+set'>quotient set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/quotient+object'>quotient</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/quotient+type'>quotient type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/induction'>induction</a></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/colimit'>colimit</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/inductive+type'>inductive type</a>, <a class='existingWikiWord' href='/nlab/show/W-type'>W-type</a>, <a class='existingWikiWord' href='/nlab/show/M-type'>M-type</a></td></tr> <tr><td style='text-align: left;'>higher <a class='existingWikiWord' href='/nlab/show/induction'>induction</a></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-limit'>higher colimit</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/higher+inductive+type'>higher inductive type</a></td></tr> <tr><td style='text-align: left;'>-</td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/truncation'>0-truncated</a> <a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-limit'>higher colimit</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/quotient+inductive+type'>quotient inductive type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/coinduction'>coinduction</a></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/limit'>limit</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/coinductive+type'>coinductive type</a></td></tr> <tr><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/preset'>preset</a></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/type'>type</a> without <a class='existingWikiWord' href='/nlab/show/identity+type'>identity types</a></td></tr> <tr><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/set'>set</a> of <a class='existingWikiWord' href='/nlab/show/truth+value'>truth values</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/subobject+classifier'>subobject classifier</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/type+of+propositions'>type of propositions</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/type'>domain of discourse</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/universe'>universe</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifier</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/type+universe'>type universe</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/modality'>modality</a></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/closure+operator'>closure operator</a>, (<a class='existingWikiWord' href='/nlab/show/idempotent+monad'>idempotent</a>) <a class='existingWikiWord' href='/nlab/show/monad'>monad</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/modal+type+theory'>modal type theory</a>, <a class='existingWikiWord' href='/nlab/show/monad+%28in+computer+science%29'>monad (in computer science)</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/linear+logic'>linear logic</a></td><td style='text-align: left;'></td><td style='text-align: left;'>(<a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+category'>symmetric</a>, <a class='existingWikiWord' href='/nlab/show/closed+monoidal+category'>closed</a>) <a class='existingWikiWord' href='/nlab/show/monoidal+category'>monoidal category</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/linear+type+theory'>linear type theory</a>/<a class='existingWikiWord' href='/nlab/show/quantum+computation'>quantum computation</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/proof+net'>proof net</a></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/string+diagram'>string diagram</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/quantum+circuit+diagram'>quantum circuit</a></td></tr> <tr><td style='text-align: left;'>(absence of) <a class='existingWikiWord' href='/nlab/show/contraction+rule'>contraction rule</a></td><td style='text-align: left;'></td><td style='text-align: left;'>(absence of) <a class='existingWikiWord' href='/nlab/show/diagonal+morphism'>diagonal</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/no-cloning+theorem'>no-cloning theorem</a></td></tr> <tr><td style='text-align: left;'></td><td style='text-align: left;'></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/synthetic+mathematics'>synthetic mathematics</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/domain+specific+embedded+programming+language'>domain specific embedded programming language</a></td></tr> </tbody></table> <p> </p> <p> </p> <p> </p> </div>  <div class="revisedby"> <p> Revision on October 17, 2022 at 05:06:01 by <a href="/nlab/author/Anonymous" style="color: #005c19">Anonymous</a> See the <a href="/nlab/history/support+object" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/15268/#Item_6">Discuss</a><span class="backintime"><a 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