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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> bracket type </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/3353/#Item_13" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="type_theory">Type theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a></strong> <a class="existingWikiWord" href="/nlab/show/metalanguage">metalanguage</a>, <a class="existingWikiWord" href="/nlab/show/practical+foundations">practical foundations</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/judgement">judgement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypothetical+judgement">hypothetical judgement</a>, <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/antecedents">antecedents</a><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> <a class="existingWikiWord" href="/nlab/show/consequent">consequent</a>, <a class="existingWikiWord" href="/nlab/show/succedents">succedents</a></li> </ul> </li> </ul> <ol> <li><a class="existingWikiWord" href="/nlab/show/type+formation+rule">type formation rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+introduction+rule">term introduction rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+elimination+rule">term elimination rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/computation+rule">computation rule</a></li> </ol> <p><strong><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent</a>, <a class="existingWikiWord" href="/nlab/show/intensional+type+theory">intensional</a>, <a class="existingWikiWord" href="/nlab/show/observational+type+theory">observational type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/calculus+of+constructions">calculus of constructions</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/syntax">syntax</a></strong> <a class="existingWikiWord" href="/nlab/show/object+language">object language</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory">theory</a>, <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>/<a class="existingWikiWord" href="/nlab/show/type">type</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definition">definition</a>/<a class="existingWikiWord" href="/nlab/show/proof">proof</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a> (<a class="existingWikiWord" href="/nlab/show/proofs+as+programs">proofs as programs</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorem">theorem</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/computational+trinitarianism">computational trinitarianism</a></strong> = <br /> <strong><a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/programs+as+proofs">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/predicate">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+morphism">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+morphisms">classifying morphisms</a> / <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <a class="existingWikiWord" href="/nlab/show/display+maps">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/introduction+rule">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/counit">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/elimination+rule">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+elimination">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/zigzag+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/zigzag+identity">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">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/%28-2%29-truncated+object">(-2)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+0">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/false">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/%28-1%29-truncated+object">(-1)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-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/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/%28-1%29-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/h-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+quantification">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/subsingletons">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+objects">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/h-propositions">h-propositions</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/existential+quantification">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/%28-1%29-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+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/%28-1%29-truncation">(-1)-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/propositional+truncation">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/n-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/diagonal+relation">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/completely+presented+set">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/0-truncated+object">0-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+2">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">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/%28infinity%2C1%29-colimit">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/0-truncated">0-truncated</a> <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">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+types">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+values">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/domain+of+discourse">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/object+classifier">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">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">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> </div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+levels">homotopy levels</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-type+theory">2-type theory</a>, <a class="existingWikiWord" href="/michaelshulman/show/2-categorical+logic">2-categorical logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory+-+contents">homotopy type theory - contents</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a>, <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>, <a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+homotopy+type+theory">directed homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HoTT+methods+for+homotopy+theorists">HoTT methods for homotopy theorists</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/semantics">semantics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>, <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/display+map">display map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+a+topos">internal logic of a topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mitchell-Benabou+language">Mitchell-Benabou language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kripke-Joyal+semantics">Kripke-Joyal semantics</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/type-theoretic+model+category">type-theoretic model category</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/type+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> <ul> <li><a href='#AsHigherInductiveType'>As a higher inductive type</a></li> <ul> <li><a href='#inference_rules'>Inference rules</a></li> </ul> <li><a href='#with_a_type_of_all_propositions'>With a type of all propositions</a></li> <li><a href='#as_localization'>As localization</a></li> <li><a href='#as_a_sequential_colimit'>As a sequential colimit</a></li> <li><a href='#as_a_quotient_set'>As a quotient set</a></li> <li><a href='#using_functions_from_the_boolean_domain'>Using functions from the boolean domain</a></li> <li><a href='#using_hubs_and_spokes'>Using hubs and spokes</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#weakly_constant_functions'>Weakly constant functions</a></li> <li><a href='#functions_from_the_interval_type'>Functions from the interval type</a></li> </ul> <li><a href='#semantics'>Semantics</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>There are various different paradigms for the interpretation of <a class="existingWikiWord" href="/nlab/show/predicate+logic">predicate logic</a> in <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>. In “logic-enriched type theory”, there is a separate class of “<a class="existingWikiWord" href="/nlab/show/propositions">propositions</a>” from the class of “<a class="existingWikiWord" href="/nlab/show/types">types</a>”. But we can also identify propositions with particular types. In the <em><a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a></em>-paradigm, every proposition is a type, and also every type is identified with a proposition (the proposition that it is an <a class="existingWikiWord" href="/nlab/show/inhabited+type">inhabited type</a>).</p> <p>By contrast, in the paradigm that may be called <a class="existingWikiWord" href="/nlab/show/propositions+as+some+types">propositions as some types</a>, every proposition is a type, but not every type is a proposition. The types which are propositions are generally those which “have at most one inhabitant” — in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> this is called being of <a class="existingWikiWord" href="/nlab/show/h-level+1">h-level 1</a> or being a <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">(-1)-type</a>. This paradigm is often used in the <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> of type theory, such as the <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> of various kinds of categories.</p> <p>Under “propositions as types”, all type-theoretic operations represent corresponding logical operations (<a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> is the <a class="existingWikiWord" href="/nlab/show/existential+quantifier">existential quantifier</a>, <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> the <a class="existingWikiWord" href="/nlab/show/universal+quantifier">universal quantifier</a>, and so on). However, under “propositions as some types”, not every such operation preserves the class of propositions; this is particularly the case for <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> and <a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a>(<a class="existingWikiWord" href="/nlab/show/or">or</a>). Thus, in order to obtain the correct logical operations, we need to reflect these constructions back into propositions somehow, finding the “underlying proposition”, corresponding to the <a class="existingWikiWord" href="/nlab/show/truncated">(-1)-truncation</a>/<a class="existingWikiWord" href="/nlab/show/h-level+1">h-level 1-projection</a>. This operation in type theory is called the <strong>bracket type</strong> (when denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A]</annotation></semantics></math>); in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> it can be identified with the <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>isInhab</mi></mrow><annotation encoding="application/x-tex">isInhab</annotation></semantics></math>.</p> <h2 id="definition">Definition</h2> <h3 id="AsHigherInductiveType">As a higher inductive type</h3> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, the propositional truncation of a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is defined as the <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a> generated by the two constructors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isinhab</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathrm{isinhab} \colon A \to [A]</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">supp</mi><mo lspace="verythinmathspace">:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>A</mi></mrow></munder><mi mathvariant="normal">isinhab</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow></msub><mi mathvariant="normal">isinhab</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{supp} \colon \prod_{x:A} \prod_{y:A} \mathrm{isinhab}(x) =_{[A]} \mathrm{isinhab}(y)</annotation></semantics></math></div> <p>In any <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> with <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a>, given type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, the <strong>support</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(A)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>isInhab</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isInhab(A)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">\tau_{-1} A</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>A</mi><msub><mo stretchy="false">‖</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\| A \|_{-1}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>A</mi><mo stretchy="false">‖</mo></mrow><annotation encoding="application/x-tex">\| A \|</annotation></semantics></math> or, lastly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A]</annotation></semantics></math>, is the <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a> defined by the two constructors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>isinhab</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> a \colon A \;\vdash \; isinhab(a) \colon supp(A) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>y</mi><mo lspace="verythinmathspace">:</mo><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mi>inpath</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> x \colon supp(A) \;,\; y \colon supp(A) \;\vdash \; inpath(x,y) \colon (x = y) \,, </annotation></semantics></math></div> <p>where in the last <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a> on the right we have the <a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>. (<a href="#Voevodsky">Voevodsky</a>, <a href="#HoTTLibrary">HoTTLibrary</a>)</p> <p>This says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(A)</annotation></semantics></math> is the type which is <a class="existingWikiWord" href="/nlab/show/universal+property">universal</a> with the property that the <a class="existingWikiWord" href="/nlab/show/terms">terms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> map to it and that any two term of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> become <a class="existingWikiWord" href="/nlab/show/equivalence+in+homotopy+type+theory">equivalent</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(A)</annotation></semantics></math>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a> <a class="existingWikiWord" href="/nlab/show/syntax">syntax</a> this is</p> <pre><code>data isinhab {i : Level} (A : Set i) : Set i where inhab : A → isinhab A inhab-path : (x y : isinhab A) → x ≡ y</code></pre> <p>The <a class="existingWikiWord" href="/nlab/show/recursion+principle">recursion principle</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(A)</annotation></semantics></math> says that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a> and we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f: A \to B</annotation></semantics></math>, then there is an induced <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">g : supp(A) \to B</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>isinhab</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(isinhab(a)) \equiv f(a)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a:A</annotation></semantics></math>. In other words, any mere proposition which follows from (the inhabitedness of) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> already follows from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(A)</annotation></semantics></math>. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">supp(A)</annotation></semantics></math>, as a mere proposition, contains no more information than the inhabitedness of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/induction+principle">induction principle</a> states that if we have a family of mere propositions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(x)</annotation></semantics></math> indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x:A</annotation></semantics></math> and we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>isinhab</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(isinhab(x))</annotation></semantics></math>, then there is an induced <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>z</mi><mo>:</mo><mi>supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g:\prod_{z:supp(A)} B(z)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>isinhab</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(isinhab(a)) \equiv f(a)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a:A</annotation></semantics></math>.</p> <h4 id="inference_rules">Inference rules</h4> <p>The inference rules for bracket types are as follows:</p> <ul> <li>Formation rules for bracket types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash [A] \; \mathrm{type}}</annotation></semantics></math></div> <ul> <li>Introduction rules for bracket types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo>:</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A \vdash [x]:[A] \; \mathrm{type}}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><msub><mi mathvariant="normal">trunc</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><msub><mo>=</mo> <mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">[</mo><mi>y</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A, y:A \vdash \mathrm{trunc}_A(x, y):[x] =_{[A]} [y] \; \mathrm{type}}</annotation></semantics></math></div> <ul> <li>Elimination rules for bracket types:</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>p</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>z</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mi>y</mi><msub><mo>=</mo> <mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>z</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash p:\prod_{x:A} \prod_{y:B(x)} \prod_{z:B(x)} y =_{B(x)} z \quad \Gamma \vdash f:\prod_{x:A} B([x])}{\Gamma, x:[A] \vdash \mathrm{ind}_{[A]}^{B(-)}(p, f, x):B(x)}</annotation></semantics></math></div> <ul> <li>Computation rules for bracket types</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi mathvariant="normal">type</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>p</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>z</mi><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mi>y</mi><msub><mo>=</mo> <mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mi>z</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>f</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>a</mi><mo>:</mo><mi>A</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msubsup><mi mathvariant="normal">ind</mi> <mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow> <mrow><mi>B</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>f</mi><mo>,</mo><mo stretchy="false">[</mo><mi>a</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>a</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash p:\prod_{x:A} \prod_{y:B(x)} \prod_{z:B(x)} y =_{B(x)} z \quad \Gamma \vdash f:\prod_{x:A} B([x]) \quad \Gamma \vdash a:A}{\Gamma \vdash \mathrm{ind}_{[A]}^{B(-)}(p, f, [a]) \equiv f(a):B([a])}</annotation></semantics></math></div> <h3 id="with_a_type_of_all_propositions">With a type of all propositions</h3> <p>Suppose the <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> has a <a class="existingWikiWord" href="/nlab/show/univalent">univalent</a> <a class="existingWikiWord" href="/nlab/show/type+of+all+propositions">type of all propositions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">Prop</mi><mo>,</mo><mi mathvariant="normal">El</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathrm{Prop}, \mathrm{El})</annotation></semantics></math>. Then the bracket type of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> could be defined as the type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>P</mi><mo>:</mo><mi mathvariant="normal">Prop</mi></mrow></munder><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi mathvariant="normal">El</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">El</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[A] \coloneqq \prod_{P:\mathrm{Prop}} (A \to \mathrm{El}(P)) \to \mathrm{El}(P)</annotation></semantics></math></div> <h3 id="as_localization">As localization</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝟙</mi></mrow><annotation encoding="application/x-tex">\mathbb{1}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝟚</mi></mrow><annotation encoding="application/x-tex">\mathbb{2}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/boolean+domain">boolean domain</a>. The bracket type of a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/localization+of+a+type+at+a+family+of+functions">localization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> at the unique function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝟚</mi><mo>→</mo><mi>𝟙</mi></mrow><annotation encoding="application/x-tex">\mathbb{2} \to \mathbb{1}</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mi>A</mi><mo>]</mo></mrow><mo>≔</mo><msub><mi>L</mi> <mi>𝟚</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\left[A\right] \coloneqq L_\mathbb{2}(A)</annotation></semantics></math></div> <p>By definition, the type of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝟙</mi><mo>→</mo><mrow><mo>[</mo><mi>A</mi><mo>]</mo></mrow><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>𝟚</mi><mo>→</mo><mrow><mo>[</mo><mi>A</mi><mo>]</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{1} \to \left[A\right]) \to (\mathbb{2} \to \left[A\right])</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a>.</p> <p>This is the special case of the <a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a> as the <a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a> is localization at the unique map from 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>-dimensional sphere type to the unit type, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝟚</mi></mrow><annotation encoding="application/x-tex">\mathbb{2}</annotation></semantics></math> is the zero-dimensional <a class="existingWikiWord" href="/nlab/show/sphere+type">sphere type</a>.</p> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></em>.</p> <h3 id="as_a_sequential_colimit">As a sequential colimit</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>*</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A * B</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/join+type">join type</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><mo>*</mo><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">A^{*(n)}</annotation></semantics></math> be the iterated join type, inductively defined on the natural numbers by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><mo>*</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup><mo>≔</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">A^{*(0)} \coloneqq \emptyset</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><mo>*</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>≔</mo><msup><mi>A</mi> <mrow><mo>*</mo><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msup><mo>*</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A^{*(n + 1)} \coloneqq A^{*(n)} * A</annotation></semantics></math>.</p> <p>Then the propositional truncation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a> of the sequence of functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><mo>*</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup><mover><mo>→</mo><mi mathvariant="normal">inr</mi></mover><msup><mi>A</mi> <mrow><mo>*</mo><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mover><mo>→</mo><mi mathvariant="normal">inr</mi></mover><msup><mi>A</mi> <mrow><mo>*</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msup><mover><mo>→</mo><mi mathvariant="normal">inr</mi></mover><mi>…</mi></mrow><annotation encoding="application/x-tex">A^{*(0)} \overset{\mathrm{inr}}\to A^{*(1)} \overset{\mathrm{inr}}\to A^{*(2)} \overset{\mathrm{inr}}\to \ldots</annotation></semantics></math></div> <h3 id="as_a_quotient_set">As a quotient set</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> be a type and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(x, y)</annotation></semantics></math> be a binary family of <a class="existingWikiWord" href="/nlab/show/contractible+types">contractible types</a> indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x:A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">y:A</annotation></semantics></math>, so that the product projection function for the dependent sum type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>:</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>z</mi><mo>:</mo><mi>A</mi><mo>×</mo><mi>A</mi></mrow></munder><mi>R</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>→</mo><mo stretchy="false">(</mo><mi>A</mi><mo>×</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_1:\left(\sum_{z:A \times A} R(\pi_1(z), \pi_2(z))\right) \to (A \times A)</annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a>. Then the propositional truncation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">A / R</annotation></semantics></math>. (The quotient set in <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> can be defined using inference rules without propositional truncations.)</p> <h3 id="using_functions_from_the_boolean_domain">Using functions from the boolean domain</h3> <p>There is another definition of the <a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> as a <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a> using functions from the boolean domain. The bracket type is generated by the two constructors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isinhab</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathrm{isinhab} \colon A \to [A]</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">supp</mi><mo lspace="verythinmathspace">:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>:</mo><mi mathvariant="normal">bool</mi><mo>→</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="normal">false</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="normal">true</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{supp} \colon \prod_{f:\mathrm{bool} \to [A]} f(\mathrm{false}) =_{[A]} f(\mathrm{true})</annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/induction+principle">induction principle</a> states that if we have a family of mere propositions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(x)</annotation></semantics></math> indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x:A</annotation></semantics></math> and we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi mathvariant="normal">isinhab</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:\prod_{x:A} B(\mathrm{isinhab}(x))</annotation></semantics></math>, then there is an induced <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>z</mi><mo>:</mo><mi mathvariant="normal">supp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g:\prod_{z:\mathrm{supp}(A)} B(z)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi mathvariant="normal">isinhab</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(\mathrm{isinhab}(a)) \equiv f(a)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a:A</annotation></semantics></math>. Here, a mere proposition is defined as a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> for which every function from the <a class="existingWikiWord" href="/nlab/show/boolean+domain">boolean domain</a> is a weakly constant function, or equivalently, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi mathvariant="normal">bool</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f:\mathrm{bool} \to A</annotation></semantics></math> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="normal">false</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>A</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="normal">true</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\mathrm{false}) =_A f(\mathrm{true})</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isProp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>:</mo><mi mathvariant="normal">bool</mi><mo>→</mo><mi>A</mi></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="normal">false</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>A</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="normal">true</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{isProp}(A) \coloneqq \prod_{f:\mathrm{bool} \to A} f(\mathrm{false}) =_A f(\mathrm{true})</annotation></semantics></math></div> <h3 id="using_hubs_and_spokes">Using hubs and spokes</h3> <p>In section 7.3 of the <a class="existingWikiWord" href="/nlab/show/HoTT+Book">HoTT Book</a> <a href="#UFP13">UFP13</a>, the authors give another definition of the propositional truncation using the hubs and spokes construction developed in section 6.7. Since the <a class="existingWikiWord" href="/nlab/show/boolean+domain">boolean domain</a> is the zero-<a class="existingWikiWord" href="/nlab/show/dimensional">dimensional</a> <a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, the propositional truncation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a> generated by</p> <ul> <li> <p>A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><mi>A</mi><mo>→</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-]:A \to [A]</annotation></semantics></math></p> </li> <li> <p>For each function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>:</mo><mi mathvariant="normal">bool</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">r:\mathrm{bool} \to A</annotation></semantics></math>, a hub point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">h(r):A</annotation></semantics></math></p> </li> <li> <p>For each function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>:</mo><mi mathvariant="normal">bool</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">r:\mathrm{bool} \to A</annotation></semantics></math> and each boolean <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi mathvariant="normal">bool</mi></mrow><annotation encoding="application/x-tex">x:\mathrm{bool}</annotation></semantics></math>, a spoke path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>r</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>A</mi></msub><mi>h</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s_r(x):r(x) =_A h(r)</annotation></semantics></math></p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="weakly_constant_functions">Weakly constant functions</h3> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, a <a class="existingWikiWord" href="/nlab/show/weakly+constant+function">weakly constant function</a> from type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A \to B</annotation></semantics></math> with a dependent function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>B</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p:\prod_{x:A} \prod_{y:A} f(x) =_B f(y)</annotation></semantics></math>.</p> <p>Given a weakly constant function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A \to B</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>B</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p:\prod_{x:A} \prod_{y:A} f(x) =_B f(y)</annotation></semantics></math>, by the recursion principle of bracket types, one has</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi mathvariant="normal">rec</mi> <mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow> <mi>B</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\mathrm{rec}_{[A]}^B(f, p):[A] \to B</annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi mathvariant="normal">rec</mi> <mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow> <mi>B</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="normal">isinhab</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\mathrm{rec}_{[A]}^B(f, p)(\mathrm{isinhab}(x)) \equiv f(x):B</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">ap</mi> <mrow><msubsup><mi mathvariant="normal">rec</mi> <mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow> <mi>B</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">supp</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≡</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>B</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{ap}_{\mathrm{rec}_{[A]}^B(f, p)}(\mathrm{supp}(x, y)) \equiv p(x, y):f(x) =_B f(y)</annotation></semantics></math></div> <p>In particular, by the judgmental <a class="existingWikiWord" href="/nlab/show/congruence+rule">congruence rule</a> for the introduction rule of <a class="existingWikiWord" href="/nlab/show/function+types">function types</a>, every weakly constant function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A \to B</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>B</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p:\prod_{x:A} \prod_{y:A} f(x) =_B f(y)</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A]</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>≡</mo><msubsup><mi mathvariant="normal">rec</mi> <mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow> <mi>B</mi></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>∘</mo><mi mathvariant="normal">isinhab</mi></mrow><annotation encoding="application/x-tex">f \equiv \mathrm{rec}_{[A]}^B(f, p) \circ \mathrm{isinhab}</annotation></semantics></math></div> <p>Weakly constant functions can also be regarded directly as functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">[A] \to B</annotation></semantics></math>, similar to how paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> can be regarded as functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕀</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbb{I} \to A</annotation></semantics></math> from the the <a class="existingWikiWord" href="/nlab/show/interval+type">interval type</a> rather than the application of said function along the path generator of the interval type.</p> <h3 id="functions_from_the_interval_type">Functions from the interval type</h3> <p>Given a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, there exists a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>A</mi><mo>→</mo><mi>𝕀</mi><mo>→</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">A \to A \to \mathbb{I} \to [A]</annotation></semantics></math> defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>λ</mi><mi>y</mi><mo>:</mo><mi>A</mi><mo>.</mo><msub><mi mathvariant="normal">rec</mi> <mi>𝕀</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi mathvariant="normal">supp</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>→</mo><mi>𝕀</mi><mo>→</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\lambda x:A.\lambda y:A.\mathrm{rec}_{\mathbb{I}}([x], [y], \mathrm{supp}(x, y)):A \to A \to \mathbb{I} \to [A]</annotation></semantics></math></div> <h2 id="semantics">Semantics</h2> <p>One presentation of the <a class="existingWikiWord" href="/nlab/show/internal+logic">internal</a> <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> of <a class="existingWikiWord" href="/nlab/show/regular+categories">regular categories</a> consists of <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> with the <a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a>, <span class="newWikiWord">strong extensional equality types<a href="/nlab/new/strong+extensional+equality+types">?</a></span>, strong <a class="existingWikiWord" href="/nlab/show/dependent+sums">dependent sums</a>, and bracket types. (The internal logic of a regular category can alternatively be presented as a <span class="newWikiWord">logic-enriched type theory<a href="/nlab/new/logic-enriched+type+theory">?</a></span>.)</p> <p>The <a class="existingWikiWord" href="/nlab/show/semantics">semantics</a> of bracket types in a <a class="existingWikiWord" href="/nlab/show/regular+category">regular category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is as follows.</p> <p>A <a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a> (a type in <a class="existingWikiWord" href="/nlab/show/context">context</a> <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>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⊢</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Type</mi></mrow><annotation encoding="application/x-tex">x\colon X \vdash A(x) \colon Type</annotation></semantics></math></div> <p>is interpreted in <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> as an arbitrary <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \\ \downarrow \\ X } \,. </annotation></semantics></math></div> <p>The corresponding bracket type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⊢</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo lspace="verythinmathspace">:</mo><mi>Type</mi></mrow><annotation encoding="application/x-tex"> x\colon X \vdash [A(x)] \colon Type </annotation></semantics></math></div> <p>is interpreted then as the <a class="existingWikiWord" href="/nlab/show/image">image</a>-<a class="existingWikiWord" href="/nlab/show/%28epi%2C+mono%29+factorization+system">factorization</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>:</mo><mo>=</mo><mi>im</mi><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &&\to&& [A] & := im(A \to X) \\ & \searrow && \swarrow \\ && X \,. } </annotation></semantics></math></div> <p>Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">[A] \to X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>, and hence the interpretation of a <a class="existingWikiWord" href="/nlab/show/proposition">proposition</a> about the elements of <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>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/image">image</a>, <a class="existingWikiWord" href="/nlab/show/inhabited+type">inhabited type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cone+type">cone type</a>, <a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a>, <a class="existingWikiWord" href="/nlab/show/set+truncation">set truncation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weakly+constant+function">weakly constant function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/propositional+truncation+object">propositional truncation object</a></p> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/homotopy+level">homotopy level</a></th><th><a class="existingWikiWord" href="/nlab/show/n-truncated+object+in+an+%28infinity%2C1%29-category">n-truncation</a></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos+theory">higher topos theory</a></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></th></tr></thead><tbody><tr><td style="text-align: left;">h-level 0</td><td style="text-align: left;">(-2)-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28-2%29-groupoid">(-2)-groupoid</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/true">true</a>/<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a>/<a class="existingWikiWord" href="/nlab/show/contractible+type">contractible type</a></td></tr> <tr><td style="text-align: left;">h-level 1</td><td style="text-align: left;">(-1)-truncated</td><td style="text-align: left;">contractible-if-<a class="existingWikiWord" href="/nlab/show/inhabited+space">inhabited</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28-1%29-groupoid">(-1)-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-sheaf">(0,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/ideal">ideal</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a>/<a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a></td></tr> <tr><td style="text-align: left;">h-level 2</td><td style="text-align: left;">0-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+0-type">homotopy 0-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/0-groupoid">0-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-set">h-set</a></td></tr> <tr><td style="text-align: left;">h-level 3</td><td style="text-align: left;">1-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+1-type">homotopy 1-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/1-groupoid">1-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/stack">stack</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-groupoid">h-groupoid</a></td></tr> <tr><td style="text-align: left;">h-level 4</td><td style="text-align: left;">2-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+2-type">homotopy 2-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></td><td style="text-align: left;">(3,1)-sheaf/2-stack</td><td style="text-align: left;">h-2-groupoid</td></tr> <tr><td style="text-align: left;">h-level 5</td><td style="text-align: left;">3-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+3-type">homotopy 3-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></td><td style="text-align: left;">(4,1)-sheaf/3-stack</td><td style="text-align: left;">h-3-groupoid</td></tr> <tr><td style="text-align: left;">h-level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n+2</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><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-truncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28n%2B1%2C1%29-sheaf">(n+1,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/n-stack">n-stack</a></td><td style="text-align: left;">h-<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>-groupoid</td></tr> <tr><td style="text-align: left;">h-level <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></td><td style="text-align: left;">untruncated</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a></td><td style="text-align: left;">h-<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>-groupoid</td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>The original articles are</p> <ul> <li>M.E. Maietti, <em>The Type Theory of Categorical Universes</em> PhD thesis, Università Degli Studi di Padova, 1998</li> </ul> <p>(which speaks of “mono types”) and</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Frank+Pfenning">Frank Pfenning</a>, <em>Intensionality, extensionality, and proof irrelevance in modal type theory</em>, In Proceedings of the 16th Annual Symposium on Logic in Computer Science (LICS’01), June 2001.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steve+Awodey">Steve Awodey</a>, <a class="existingWikiWord" href="/nlab/show/Andrej+Bauer">Andrej Bauer</a>, <em>Propositions as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math>Types<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></em>, Journal of Logic and Computation, <strong>14</strong> (2004) 447-471 [<a href="https://doi.org/10.1093/logcom/14.4.447">doi:10.1093/logcom/14.4.447</a>, <a href="http://andrej.com/papers/brackets_letter.pdf">pdf</a>]</p> </li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li id="UFP13"><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, §3.7 <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em> (2013) [<a href="http://homotopytypetheory.org/book/">web</a>, <a href="http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf">pdf</a>]</li> </ul> <p>Exposition:</p> <ul> <li id="Shulman"><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>Minicourse on homotopy type theory</em> (2012) (<a href="http://www.sandiego.edu/~shulman/hottminicourse2012/">web</a>)</li> </ul> <p>Formalization:</p> <ul id="HoTTLibrary"> <li id="Voevodsky"> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Voevodsky">Vladimir Voevodsky</a>, <em>The hProp version of the “inhabited” construction.</em> (<a href="http://www.math.ias.edu/~vladimir/Foundations_library/hProp.html#lab140">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Coq">Coq</a><a href="https://github.com/HoTT/HoTT/tree/master/Coq">HoTT library</a>, <em><a href="https://github.com/HoTT/HoTT/blob/master/Coq/HIT/IsInhab.v">IsInhab.v</a></em></p> </li> </ul> <p>Discussion in the more general context of <a class="existingWikiWord" href="/nlab/show/n-truncation+modality"><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>-truncations</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Guillaume+Brunerie">Guillaume Brunerie</a>, <em>Truncations and truncated higher inductive types</em> (<a href="http://homotopytypetheory.org/2012/09/16/truncations-and-truncated-higher-inductive-types/">web</a>)</li> </ul> <p>More on the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of propositional truncation:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicolai+Kraus">Nicolai Kraus</a>, <em>The General Universal Property of the Propositional Truncation</em>, in <em>TYPES 2014</em> Leibniz International Proceedings in Informatics (LIPIcs) <strong>39</strong> (2015) [<a href="https://arxiv.org/abs/1411.2682">arXiv:1411.2682</a>, <a href="https://doi.org/10.4230/LIPIcs.TYPES.2014.111">doi:10.4230/LIPIcs.TYPES.2014.111</a>]</li> </ul> <p>For propositional truncations as sequential colimits, see section 26.5 of</p> <ul> <li id="RijkeDraft22"><a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a> (2022), <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Type+Theory">Introduction to Homotopy Type Theory</a></em>, draft. (<a href="https://raw.githubusercontent.com/martinescardo/HoTTEST-Summer-School/main/HoTT/hott-intro.pdf">pdf</a>)</li> </ul> <p>as well as section 16.2 of the lecture notes:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, <em>The homotopy image of a map</em>, Lecture 16 in: <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Type+Theory">Introduction to Homotopy Type Theory</a></em>, lecture notes, CMU (2018) [<a href="http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Rijke-IntroductionHoTT-2018.pdf" title="pdf">pdf</a>, <a href="https://www.andrew.cmu.edu/user/erijke/hott/">webpage</a>]</li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/n-truncations">n-truncations</a> as <a class="existingWikiWord" href="/nlab/show/localizations">localizations</a> at <a class="existingWikiWord" href="/nlab/show/sphere+types">sphere types</a>, see:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory (<a href="https://arxiv.org/abs/2205.15887">arXiv:2205.15887</a>)</li> </ul> <p>That propositional truncations with judgmental computation rules along with the <a class="existingWikiWord" href="/nlab/show/boolean+domain">boolean domain</a> imply function extensionality:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicolai+Kraus">Nicolai Kraus</a>, <a class="existingWikiWord" href="/nlab/show/Mart%C3%ADn+Escard%C3%B3">Martín Escardó</a>, <a class="existingWikiWord" href="/nlab/show/Thierry+Coquand">Thierry Coquand</a>, <a class="existingWikiWord" href="/nlab/show/Thorsten+Altenkirch">Thorsten Altenkirch</a>, <em>Notions of Anonymous Existence in Martin-Löf Type Theory</em>, Logical Methods in Computer Science <strong>13</strong> 1 [<a href="https://doi.org/10.23638/LMCS-13(1:15)2017">doi:10.23638/LMCS-13(1:15)2017</a>, <a href="https://arxiv.org/abs/1610.03346">arXiv:1610.03346</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 6, 2024 at 11:37:49. 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