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12 2 1.1 11 6.9 11.4 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> essentially algebraic theory </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/6466/#Item_11" 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="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> <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> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</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>A <a class="existingWikiWord" href="/nlab/show/mathematical+structure">mathematical structure</a> is <em>essentially algebraic</em> if its definition involves <a class="existingWikiWord" href="/nlab/show/functional+relation">partially defined operations</a> satisfying equational laws, where the <a class="existingWikiWord" href="/nlab/show/domain">domain</a> of any given operation is a subset where various other operations happen to be equal. An actual <a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> is one where all operations are total <a class="existingWikiWord" href="/nlab/show/function">function</a>s.</p> <p>The most familiar example may be the (<a class="existingWikiWord" href="/nlab/show/strict+category">strict</a>) notion of <a class="existingWikiWord" href="/nlab/show/category">category</a>: a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> consists of a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">C_0</annotation></semantics></math> of objects, a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math> of morphisms, source and target maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>:</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">s,t : C_1 \to C_0</annotation></semantics></math> and so on, but composition is only defined for pairs of morphisms where the source of one happens to equal the target of the other.</p> <p>Essentially algebraic theories can be understood through <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> at least when they are finitary, so that all operations have only finitely many arguments. This gives a generalisation of <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theories</a>, which describe finitary <a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theories</a>.</p> <p>As the domains of the operations are given by the solutions to equations, they may be understood using the notion of <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>. So, just as a Lawvere theory is defined using a category with finite <a class="existingWikiWord" href="/nlab/show/product">product</a>s, a finitary essentially algebraic theory is defined using a category with <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> — or in other words, finite products and also equalizers (from which all other finite limits, including <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>s, may be derived).</p> <h2 id="definition">Definition</h2> <p>As alluded to above, the most concise and elegant definition is through category theory. The traditional definition is this:</p> <div class="un_def"> <h6 id="definition_2">Definition</h6> <p>An <strong>essentially algebraic theory</strong> or <strong>finite limits theory</strong> is a category that is <a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finitely complete</a>, i.e., has all finite limits. A <strong>model</strong> of an essentially algebraic theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>T</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F: T \to Set</annotation></semantics></math></div> <p>that is <a class="existingWikiWord" href="/nlab/show/left+exact">left exact</a>, i.e., preserves all finite limits. A <strong>homomorphism</strong> of models is a natural transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>F</mi><mo>→</mo><mi>F</mi><mo>′</mo></mrow><annotation encoding="application/x-tex"> \alpha : F \to F'</annotation></semantics></math></div> <p>between left exact functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>,</mo><mi>F</mi><mo>′</mo><mo>:</mo><mi>T</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F, F' : T \to Set</annotation></semantics></math>. Models of an essentially algebraic theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> and the homomorphisms between them form a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Lex</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mod(T) = Lex(T, Set)</annotation></semantics></math>.</p> <p>More generally, for any category with finite limits <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>, we can define the category of <strong>models of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in <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></strong>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lex</mi><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Lex(T, X)</annotation></semantics></math>, which has left exact functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>T</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">F: T \to X</annotation></semantics></math> as objects and natural transformations between these as morphisms.</p> </div> <p>However, the finiteness restriction on the limits above is not part of the core concept of an ‘essentially algebraic’ structure, so one may prefer to call a category with finite limits a <strong>finitary</strong> essentially algebraic theory. We do this in what follows.</p> <p>A more traditional syntactic definition (following Adámek–Rosicky; see the references below) is as follows:</p> <div class="un_def"> <h6 id="definition_3">Definition</h6> <p>An <strong>essentially algebraic theory</strong> is a quadruple</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>E</mi><mo>,</mo><msub><mi>Σ</mi> <mi>t</mi></msub><mo>,</mo><mi>Def</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma = (\Sigma, E, \Sigma_t, Def)</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is a many-sorted <a class="existingWikiWord" href="/nlab/show/signature+%28in+logic%29">signature</a> consisting only of operation symbols, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>-equations, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>t</mi></msub><mo>⊆</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma_t \subseteq \Sigma</annotation></semantics></math> is a set of operation symbols called “total”, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Def</mi></mrow><annotation encoding="application/x-tex">Def</annotation></semantics></math> is a function which assigns, to each operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><mi>Σ</mi><mo>−</mo><msub><mi>Σ</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\sigma \in \Sigma - \Sigma_t</annotation></semantics></math> of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>s</mi> <mi>i</mi></msub><mo>→</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">\prod_{i \in I} s_i \to s</annotation></semantics></math>, a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Def</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Def(\sigma)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_t</annotation></semantics></math>-equations involving only variables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub><mo>∈</mo><mi>Var</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_i \in Var(s_i)</annotation></semantics></math>.</p> <p>A (set-theoretic) <strong>model</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> of a theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> assigns to each sort <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math> a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(s)</annotation></semantics></math>, to each operation symbol <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>s</mi> <mi>i</mi></msub><mo>→</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">\sigma: \prod_{i \in I} s_i \to s</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> a <em>partial</em> function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>M</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>M</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(\sigma): \prod_{i \in I} M(s_i) \to M(s)</annotation></semantics></math></div> <p>such that</p> <ul> <li> <p>For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><msub><mi>Σ</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\sigma \in \Sigma_t</annotation></semantics></math> the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(\sigma)</annotation></semantics></math> is a total function;</p> </li> <li> <p>For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><mi>Σ</mi><mo>−</mo><msub><mi>Σ</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\sigma \in \Sigma - \Sigma_t</annotation></semantics></math> of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>s</mi> <mi>i</mi></msub><mo>→</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">\prod_{i \in I} s_i \to s</annotation></semantics></math>, and each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>-tuple</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>=</mo><mo stretchy="false">⟨</mo><msub><mi>a</mi> <mi>i</mi></msub><msub><mo stretchy="false">⟩</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo>∈</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>M</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">a = \langle a_i \rangle_{i \in I} \in \prod_{i \in I} M(s_i),</annotation></semantics></math></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(\sigma)(a)</annotation></semantics></math> is defined if and only if all the equations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Def</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Def(\sigma)</annotation></semantics></math> are satisfied at the argument <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> </li> <li> <p>All the equations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> are satisfied (i.e. the restrictions of both partial functions to the intersection of their domains of definitions are equal).</p> </li> </ul> </div> <p>Homomorphisms of models <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>M</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\theta: M \to M'</annotation></semantics></math> are defined in the standard way: a collection of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>:</mo><mi>M</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>→</mo><mi>M</mi><mo>′</mo><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\theta(s): M(s) \to M'(s)</annotation></semantics></math> for each sort of the signature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> which are compatible with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>M</mi><mo>′</mo><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(\sigma), M'(\sigma)</annotation></semantics></math> in the evident way.</p> <h2 id="properties">Properties</h2> <p>The point is that (in the finitary case) either notion of theory may be used to specify the same category of models, and that</p> <div class="standout"> <p>Categories of models of finitary essentially algebraic theories are precisely equivalent to <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally finitely presentable categories</a>. These are equivalent to categories of models of finite limit <a class="existingWikiWord" href="/nlab/show/sketch">sketch</a>es.</p> </div> <p>A <a class="existingWikiWord" href="/nlab/show/duality">duality</a> between essentially algebraic theories and their categories of models is given by <a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>A (multisorted) <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is the same thing (has the same models) as a finitary essentially algebraic theory in which all operations are total. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub></mrow><annotation encoding="application/x-tex">C_T</annotation></semantics></math> is the opposite of the category of finitely presented <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebras, then the category of models is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lex</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>T</mi></msub><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Lex(C_T, Set)</annotation></semantics></math>.</p> </li> <li> <p>As mentioned above, categories are models of a finitary essentially algebraic theory.</p> </li> <li> <p>An equational <a class="existingWikiWord" href="/nlab/show/Horn+theory">Horn theory</a> is essentially algebraic, but not all essentially algebraic theories are equational Horn theories. Perhaps surprisingly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> is not the category of models of any equational Horn theory, nor is even the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pos</mi></mrow><annotation encoding="application/x-tex">Pos</annotation></semantics></math> of posets. See <a href="ftp://132.206.150.195/pub/barr/pdffiles/horn.pdf">this paper</a> by Barr for a proof. Essentially algebraic theories are equivalent to <em>partial</em> Horn theories (<a href="#PalmgrenVickers">Palmgren, Vickers</a>).</p> </li> <li> <p>An equivalent formulation is as a <em>cartesian theory</em>, a <a class="existingWikiWord" href="/nlab/show/geometric+theory">geometric theory</a> in which disjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo></mrow><annotation encoding="application/x-tex">\bigvee</annotation></semantics></math> is not used, and each use of existential quantification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∃</mo></mrow><annotation encoding="application/x-tex">\exists</annotation></semantics></math> must be accompanied by a proof that existence is unique. See <a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> / <strong>essentially algebraic theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-Lawvere+theory">2-Lawvere theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+%28%E2%88%9E%2C1%29-theory">algebraic (∞,1)-theory</a> / <a class="existingWikiWord" href="/nlab/show/essentially+algebraic+%28%E2%88%9E%2C1%29-theory">essentially algebraic (∞,1)-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+logic">cartesian logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+algebraic+theory">generalized algebraic theory</a></p> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/category">category</a></th><th><a class="existingWikiWord" href="/nlab/show/functor">functor</a></th><th><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></th><th><a class="existingWikiWord" href="/nlab/show/theory">theory</a></th><th><a class="existingWikiWord" href="/nlab/show/hyperdoctrine">hyperdoctrine</a></th><th><a class="existingWikiWord" href="/nlab/show/subobject+poset">subobject poset</a></th><th><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a></th><th><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finitely complete category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+functor">cartesian functor</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+logic">cartesian logic</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/essentially+algebraic+theory">essentially algebraic theory</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/lextensive+category">lextensive category</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjunctive+logic">disjunctive logic</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/regular+category">regular category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/regular+functor">regular functor</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/regular+logic">regular logic</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/regular+theory">regular theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/regular+hyperdoctrine">regular hyperdoctrine</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/infimum">infimum</a>-<a class="existingWikiWord" href="/nlab/show/semilattice">semilattice</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/regular+coverage">regular coverage</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/regular+topos">regular topos</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coherent+category">coherent category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coherent+functor">coherent functor</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coherent+logic">coherent logic</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coherent+theory">coherent theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coherent+hyperdoctrine">coherent hyperdoctrine</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coherent+coverage">coherent coverage</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coherent+topos">coherent topos</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+category">geometric category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+functor">geometric functor</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+logic">geometric logic</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+theory">geometric theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+hyperdoctrine">geometric hyperdoctrine</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/frame">frame</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+coverage">geometric coverage</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Heyting+category">Heyting category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Heyting+functor">Heyting functor</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic</a> <a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a></td><td style="text-align: left;">intuitionistic <a class="existingWikiWord" href="/nlab/show/first-order+theory">first-order theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/first-order+hyperdoctrine">first-order hyperdoctrine</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/De+Morgan+Heyting+category">De Morgan Heyting category</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic</a> <a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a> with <a class="existingWikiWord" href="/nlab/show/weak+excluded+middle">weak excluded middle</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/De+Morgan+Heyting+algebra">De Morgan Heyting algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Boolean+category">Boolean category</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/classical+logic">classical</a> <a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a></td><td style="text-align: left;">classical <a class="existingWikiWord" href="/nlab/show/first-order+theory">first-order theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Boolean+hyperdoctrine">Boolean hyperdoctrine</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/multiplicative+classical+linear+logic">multiplicative classical linear logic</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+closed+category">symmetric monoidal closed category</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/multiplicative+intuitionistic+linear+logic">multiplicative intuitionistic linear logic</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fragment">fragment</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>&</mi><mo>,</mo><mo>⊤</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\&, \top\}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cocartesian+monoidal+category">cocartesian monoidal category</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fragment">fragment</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>⊕</mo><mo>,</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\oplus, 0\}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/simply+typed+lambda+calculus">simply typed lambda calculus</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="References">References</h2> <p>Freyd first introduced essentially algebraic theories here:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Freyd">Peter Freyd</a>, <em>Aspects of Topoi</em>, Bull. Austr. Math. Soc. 7, pp. 1–76, 467–80. 1972 (<a href="https://www.cambridge.org/core/services/aop-cambridge-core/content/view/B65FB15DCD85C816F31D9C87D355AD24/S0004972700044828a.pdf/aspects-of-topoi.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Ad%C3%A1mek">Jiří Adámek</a>, M. Hébert, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, <em>On essentially algebraic theories and their generalizations</em>, Algebra Universalis, August 1999, Volume 41, Issue 3, pp 213-227</p> </li> <li id="AdamekRosicky"> <p><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Ad%C3%A1mek">Jiří Adámek</a>, <a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Rosick%C3%BD">Jiří Rosický</a>, section 3.D of <em><a class="existingWikiWord" href="/nlab/show/Locally+presentable+and+accessible+categories">Locally presentable and accessible categories</a></em>, Cambridge University Press, (1994)</p> </li> </ul> <p>A nice equivalent formulation can be found in</p> <ul> <li id="PalmgrenVickers"><a class="existingWikiWord" href="/nlab/show/Erik+Palmgren">Erik Palmgren</a>, <a class="existingWikiWord" href="/nlab/show/Steve+Vickers">Steve Vickers</a> Partial Horn logic and cartesian categories. Annals of Pure and Applied Logic, 145(2007), 314-355. (<a href="http://www2.math.uu.se/~palmgren/partialalgebras_pre.pdf">pdf</a>)</li> </ul> <p>Cartesian theories were introduced under different names in the early seventies by <a class="existingWikiWord" href="/nlab/show/John+Isbell">John Isbell</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Freyd">Peter Freyd</a> and <a class="existingWikiWord" href="/nlab/show/Michel+Coste">Michel Coste</a> (cf. <a href="#Johnstone79">Johnstone 1979</a>). A standard source is Johnstone (<a href="#Johnstone02">2002</a>).</p> <ul> <li id="Freyd02"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Freyd">Peter Freyd</a>, <em>Cartesian Logic</em> , Theor. Comp. Sci. <strong>278</strong> (2002) pp.3-21.</p> </li> <li id="Johnstone79"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>A Syntactic Approach to Diers’ Localizable Categories</em> , pp.466-478 in Springer LNM <strong>753</strong> Heidelberg 1979.</p> </li> <li id="Johnstone02"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Sketches of an <a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a> II</em> , Oxford UP 2002. (Around D1.3.4 p.833)</p> </li> </ul> <p>A short introductory/overview note:</p> <ul> <li id="keml-diagrams"><a class="existingWikiWord" href="/nlab/show/Andreas+Nuyts">Andreas Nuyts</a>, <em>Understanding Universal Algebra Using Kleisli-Eilenberg-Moore-Lawvere Diagrams</em>, <a href="https://anuyts.github.io/files/keml-diagrams.pdf">note</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 12, 2024 at 15:07:29. 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