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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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="categorical_algebra">Categorical algebra</h4> <div class="hide"><div> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>+<a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/internalization">internalization</a> and <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+object">group object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+object">ring object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+object">algebra object</a> (associative, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+object">Lie</a>, …)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+object">module object</a>/<a class="existingWikiWord" href="/nlab/show/action+object">action object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+locale">internal locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+category">internal category</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">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internal+infinity-categories+contents">more</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+site">internal site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+diagram">internal diagram</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+Lawvere+theory">algebras over</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/algebraic+theories">algebraic theories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebras over</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/monads">monads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebras over</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/operads">operads</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>, <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+category+theory+and+type+theory">relation between category theory and type theory</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#over_singlecoloured_operads'>Over single-coloured operads</a></li> <li><a href='#over_coloured_operads'>Over coloured operads</a></li> </ul> <li><a href='#literature'>Literature</a></li> <ul> <li><a href='#related_lab_entries'>Related <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>Lab entries</a></li> <li><a href='#generalizations'>Generalizations</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>An <a class="existingWikiWord" href="/nlab/show/operad">operad</a> is a structure whose elements are <em>formal operations</em>, closed under the operation of plugging some formal operations into others. An <strong>algebra over an operad</strong> is a structure in which the formal operations are interpreted as actual operations on an object, via a suitable <a class="existingWikiWord" href="/nlab/show/action">action</a>.</p> <p>Accordingly, there is a notion of <a class="existingWikiWord" href="/nlab/show/module+over+an+algebra+over+an+operad">module over an algebra over an operad</a>.</p> <h2 id="definition">Definition</h2> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed symmetric monoidal category</a> with monoidal unit <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>, and let <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> be any object. There is a canonical or tautological <a class="existingWikiWord" href="/nlab/show/operad">operad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Op(X)</annotation></semantics></math> whose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>n</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">n^{th}</annotation></semantics></math> component is the internal hom <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(X^{\otimes n}, X)</annotation></semantics></math>; the operad identity is the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>X</mi></msub><mo>:</mo><mi>I</mi><mo>→</mo><mi>M</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1_X: I \to M(X, X)</annotation></semantics></math></div> <p>and the operad multiplication is given by the composite</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>M</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><mo>⊗</mo><mi>k</mi></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>M</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><mo>⊗</mo><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>…</mi><mo>⊗</mo><mi>M</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><mo>⊗</mo><msub><mi>n</mi> <mi>k</mi></msub></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mn>1</mn><mo>⊗</mo><msub><mi>func</mi> <mo>⊗</mo></msub></mrow></mover></mtd> <mtd><mi>M</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><mo>⊗</mo><mi>k</mi></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>M</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><mo>⊗</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><mi>…</mi><mo>+</mo><msub><mi>n</mi> <mi>k</mi></msub></mrow></msup><mo>,</mo><msup><mi>X</mi> <mrow><mo>⊗</mo><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mo>→</mo><mi>comp</mi></mover></mtd> <mtd><mi>M</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><mo>⊗</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><mi>…</mi><mo>+</mo><msub><mi>n</mi> <mi>k</mi></msub></mrow></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ M(X^{\otimes k}, X) \otimes M(X^{\otimes n_1}, X) \otimes \ldots \otimes M(X^{\otimes n_k}, X) & \stackrel{1 \otimes func_\otimes}{\to} & M(X^{\otimes k}, X) \otimes M(X^{\otimes n_1 + \ldots + n_k}, X^{\otimes k}) \\ & \stackrel{comp}{\to} & M(X^{\otimes n_1 + \ldots + n_k}, X) } </annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math> be any operad in <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>. An <strong>algebra over</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math> is an object <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> equipped with an operad map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi><mo>:</mo><mi>O</mi><mo>→</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\xi: O \to Op(X)</annotation></semantics></math>. Alternatively, the data of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math>-algebra is given by a sequence of maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>X</mi> <mrow><mo>⊗</mo><mi>k</mi></mrow></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">O(k) \otimes X^{\otimes k} \to X</annotation></semantics></math></div> <p>which specifies an action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math> via finitary operations on <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>, with compatibility conditions between the operad multiplication and the structure of plugging in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> finitary operations on <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> into a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ary operation (and compatibility with actions by permutations).</p> <p>An <em>algebra over an <a class="existingWikiWord" href="/nlab/show/operad">operad</a></em> can equivalently be defined as a <a class="existingWikiWord" href="/nlab/show/category+over+an+operad">category over an operad</a> which has a single <a class="existingWikiWord" href="/nlab/show/object">object</a>.</p> <p>If <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> is cocomplete, then an operad in <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> may be defined as a monoid in the symmetric monoidal category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>M</mi> <mrow><msup><mi>ℙ</mi> <mi>op</mi></msup></mrow></msup><mo>,</mo><mo>∘</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(M^{\mathbb{P}^{op}}, \circ)</annotation></semantics></math> of permutation representations in <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>, aka <a class="existingWikiWord" href="/nlab/show/species">species</a> in <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>, with respect to the substitution product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math>. There is an <a class="existingWikiWord" href="/nlab/show/actegory">actegory</a> structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mrow><msup><mi>ℙ</mi> <mi>op</mi></msup></mrow></msup><mo>×</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">M^{\mathbb{P}^{op}} \times M \to M</annotation></semantics></math> which arises by restriction of the monoidal product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math> if we consider <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> as fully embedded in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mrow><msup><mi>ℙ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">M^{\mathbb{P}^{op}}</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>i</mi><mo>:</mo><mi>M</mi><mo>→</mo><msup><mi>M</mi> <mrow><msup><mi>ℙ</mi> <mi>op</mi></msup></mrow></msup><mo>:</mo><mi>X</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>n</mi><mo>↦</mo><msub><mi>δ</mi> <mrow><mi>n</mi><mn>0</mn></mrow></msub><mo>⋅</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i: M \to M^{\mathbb{P}^{op}}: X \mapsto (n \mapsto \delta_{n 0} \cdot X)</annotation></semantics></math></div> <p>(interpret <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> as concentrated in the 0-ary or “constants” component), so that an operad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math> induces a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>O</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{O}</annotation></semantics></math> on <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> via the actegory structure. As a functor, the monad may be defined by a <a class="existingWikiWord" href="/nlab/show/coend">coend</a> formula</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>O</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∫</mo> <mrow><mi>k</mi><mo>∈</mo><mi>ℙ</mi></mrow></msup><mi>O</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>X</mi> <mrow><mo>⊗</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\hat{O}(X) = \int^{k \in \mathbb{P}} O(k) \otimes X^{\otimes k}</annotation></semantics></math></div> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math>-algebra is the same thing as an algebra over the monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>O</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat{O}</annotation></semantics></math>.</p> <p><strong>Remark</strong> If <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 the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/enriched+category">enriching category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math> the <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>-enriched operad in question, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in Obj(C)</annotation></semantics></math> is the single <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a> of the <a class="existingWikiWord" href="/nlab/show/category+over+an+operad">O-category</a> with single object, it makes sense to write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}A</annotation></semantics></math> for that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math>-category. Compare the discussion at <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> and <a class="existingWikiWord" href="/nlab/show/group">group</a>, which are special cases of this.</p> <h2 id="examples">Examples</h2> <h3 id="over_singlecoloured_operads">Over single-coloured operads</h3> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> is an algebra over the <a class="existingWikiWord" href="/nlab/show/associative+operad">associative operad</a>.</p> <ul> <li>an <a class="existingWikiWord" href="/nlab/show/A-infinity+algebra">A-infinity algebra</a> is an algebra over a cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Assoc</mi></mrow><annotation encoding="application/x-tex">Assoc</annotation></semantics></math>.</li> </ul> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a> is an algebra over the <a class="existingWikiWord" href="/nlab/show/commutative+operad">commutative operad</a>.</p> <ul> <li>an <a class="existingWikiWord" href="/nlab/show/E-infinity+algebra">E-infinity algebra</a> is an algebra over a cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> of the commutative operad</li> </ul> </li> <li> <p>etc.</p> </li> </ul> <h3 id="over_coloured_operads">Over coloured operads</h3> <ul> <li> <p>There is a <a class="existingWikiWord" href="/nlab/show/coloured+operad">coloured operad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>P</mi></msub></mrow><annotation encoding="application/x-tex">Mod_P</annotation></semantics></math> whose algebras are pairs consisting of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>-algebra <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 a <a class="existingWikiWord" href="/nlab/show/module">module</a> over <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>For a single-coloured operad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> there is a coloured operad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">P^1</annotation></semantics></math> whose algebras are triples consisting of two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> algebras and a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_1 \to A_2</annotation></semantics></math> between them.</p> </li> <li> <p>Let <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> be a set. There is 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>-coloured operad whose algebras are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> with <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 their set of objects.</p> </li> </ul> <h2 id="literature">Literature</h2> <h3 id="related_lab_entries">Related <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>Lab entries</h3> <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> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a> / <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a></li> </ul> </li> <li> <p><strong>algebra over an operad</strong></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> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></li> </ul> </li> </ul> <h3 id="generalizations">Generalizations</h3> <ul> <li>S. N. Tronin, <em>Algebras over multicategories</em>, Russ Math. (2016) 60: 52. <a href="http://dx.doi.org/10.3103/S1066369X16020092">doi</a>; Rus. original: С. Н. Тронин, <em>Об алгебрах над мультикатегориями</em>, Изв. вузов. Матем., 2016, № 2, 62–74</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 22, 2021 at 08:27:11. 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