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microcosm principle

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Higher algebra

</head> <body> <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></body></html></div> <h4 id="higher_category_theory">Higher category theory</h4> <div class="hide"><html xml:lang='en' xmlns:svg='http://www.w3.org/2000/svg' xmlns='http://www.w3.org/1999/xhtml'> <head><meta content='application/xhtml+xml;charset=utf-8' http-equiv='Content-type' /><title /></head> <body> <p><strong><a class='existingWikiWord' href='/nlab/show/higher+category+theory'>higher category theory</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/category+theory'>category theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a></li> </ul> <h2 id='basic_concepts'>Basic concepts</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/k-morphism'>k-morphism</a>, <a class='existingWikiWord' href='/nlab/show/coherence'>coherence</a></li> <li><a class='existingWikiWord' href='/nlab/show/looping+and+delooping'>looping and delooping</a></li> <li><a class='existingWikiWord' href='/nlab/show/stabilization'>looping and suspension</a></li> </ul> <h2 id='basic_theorems'>Basic theorems</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</li> <li><a class='existingWikiWord' href='/nlab/show/delooping+hypothesis'>delooping hypothesis</a>-theorem</li> <li><a class='existingWikiWord' href='/nlab/show/periodic+table'>periodic table</a></li> <li><a class='existingWikiWord' href='/nlab/show/stabilization+hypothesis'>stabilization hypothesis</a>-theorem</li> <li><a class='existingWikiWord' href='/michaelshulman/show/exactness+hypothesis'>exactness hypothesis</a></li> <li><a class='existingWikiWord' href='/nlab/show/holographic+principle+of+higher+category+theory'>holographic principle</a></li> </ul> <h2 id='applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/higher+category+theory+and+physics'>higher category theory and physics</a></li> </ul> <h2 id='models'>Models</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/%28n%2Cr%29-category'>(n,r)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/Theta-space'>Theta-space</a></li> <li><a class='existingWikiWord' href='/nlab/show/%E2%88%9E-category'>∞-category</a>/<a class='existingWikiWord' href='/nlab/show/%E2%88%9E-category'>∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2Cn%29-category'>(∞,n)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/n-fold+complete+Segal+space'>n-fold complete Segal space</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C2%29-category'>(∞,2)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-category'>(∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/quasi-category'>quasi-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/algebraic+quasi-category'>algebraic quasi-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/simplicially+enriched+category'>simplicially enriched category</a></li> <li><a class='existingWikiWord' href='/nlab/show/complete+Segal+space'>complete Segal space</a></li> <li><a class='existingWikiWord' href='/nlab/show/model+category'>model category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C0%29-category'>(∞,0)-category</a>/<a class='existingWikiWord' href='/nlab/show/%E2%88%9E-groupoid'>∞-groupoid</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/Kan+complex'>Kan complex</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/algebraic+Kan+complex'>algebraic Kan complex</a></li> <li><a class='existingWikiWord' href='/nlab/show/simplicial+T-complex'>simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/n-category'>n-category</a> = (n,n)-category <ul> <li><a class='existingWikiWord' href='/nlab/show/2-category'>2-category</a>, <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-category'>(2,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/1-category'>1-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/0-category'>0-category</a></li> <li><span class='newWikiWord'>(?1)-category<a href='/nlab/new/%28%3F1%29-category'>?</a></span></li> <li><span class='newWikiWord'>(?2)-category<a href='/nlab/new/%28%3F2%29-category'>?</a></span></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/n-poset'>n-poset</a> = <a class='existingWikiWord' href='/nlab/show/n-poset'>(n-1,n)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/poset'>poset</a> = <a class='existingWikiWord' href='/nlab/show/%280%2C1%29-category'>(0,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/2-poset'>2-poset</a> = <a class='existingWikiWord' href='/nlab/show/%281%2C2%29-category'>(1,2)-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/n-groupoid'>n-groupoid</a> = (n,0)-category <ul> <li><a class='existingWikiWord' href='/nlab/show/2-groupoid'>2-groupoid</a>, <a class='existingWikiWord' href='/nlab/show/3-groupoid'>3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/categorification'>categorification</a>/<a class='existingWikiWord' href='/nlab/show/decategorification'>decategorification</a></li> <li><a class='existingWikiWord' href='/nlab/show/geometric+definition+of+higher+category'>geometric definition of higher category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/Kan+complex'>Kan complex</a></li> <li><a class='existingWikiWord' href='/nlab/show/quasi-category'>quasi-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories'>simplicial model for weak ∞-categories</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/complicial+set'>complicial set</a></li> <li><a class='existingWikiWord' href='/nlab/show/weak+complicial+set'>weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/algebraic+definition+of+higher+category'>algebraic definition of higher category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/bicategory'>bicategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/bigroupoid'>bigroupoid</a></li> <li><a class='existingWikiWord' href='/nlab/show/tricategory'>tricategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/tetracategory'>tetracategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/strict+%E2%88%9E-category'>strict ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/Batanin+%E2%88%9E-category'>Batanin ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/Trimble+n-category'>Trimble ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories'>Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/stable+homotopy+theory'>stable homotopy theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+category'>symmetric monoidal category</a></li> <li><a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category'>symmetric monoidal (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/stable+%28%E2%88%9E%2C1%29-category'>stable (∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/dg-category'>dg-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/A-%E2%88%9E+category'>A-∞ category</a></li> <li><a class='existingWikiWord' href='/nlab/show/triangulated+category'>triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id='morphisms'>Morphisms</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/k-morphism'>k-morphism</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/2-morphism'>2-morphism</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/transfor'>transfor</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/natural+transformation'>natural transformation</a></li> <li><a class='existingWikiWord' href='/nlab/show/modification'>modification</a></li> </ul> </li> </ul> <h2 id='functors'>Functors</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/functor'>functor</a></li> <li><a class='existingWikiWord' href='/nlab/show/2-functor'>2-functor</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/pseudofunctor'>pseudofunctor</a></li> <li><a class='existingWikiWord' href='/nlab/show/lax+functor'>lax functor</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-functor'>(∞,1)-functor</a></li> </ul> <h2 id='universal_constructions'>Universal constructions</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/2-limit'>2-limit</a></li> <li><a class='existingWikiWord' href='/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor'>(∞,1)-adjunction</a></li> <li><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension'>(∞,1)-Kan extension</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/limit+in+a+quasi-category'>(∞,1)-limit</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a></li> </ul> <h2 id='extra_properties_and_structure'>Extra properties and structure</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/cosmic+cube'>cosmic cube</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/k-tuply+monoidal+n-category'>k-tuply monoidal n-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/strict+%E2%88%9E-category'>strict ∞-category</a>, <a class='existingWikiWord' href='/nlab/show/strict+%E2%88%9E-groupoid'>strict ∞-groupoid</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/stable+%28%E2%88%9E%2C1%29-category'>stable (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-topos'>(∞,1)-topos</a></li> </ul> <h2 id='1categorical_presentations'>1-categorical presentations</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/homotopical+category'>homotopical category</a></li> <li><a class='existingWikiWord' href='/nlab/show/model+category'>model category theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/enriched+category+theory'>enriched category theory</a></li> </ul> <div> <p><a href='/nlab/edit/higher+category+theory+-+contents'>Edit this sidebar</a></p> </div></body></html></div> </div> </div> <h1 id="contents">Contents</h1> <div class="maruku_toc"><ul><li><a href="#idea">Idea</a></li><li><a href="#examples">Examples</a></li><li><a href="#formalizations">Formalizations</a><ul><li><a href="#for_categorical_operad_algebras">For <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-categorical operad algebras</a></li><li><a href="#for_categorical_algebras_over_cartesian_monads">For categorical algebras over cartesian monads</a></li><li><a href="#for_categorical_algebras_over_lawvere_theories">For categorical algebras over Lawvere theories</a></li><li><a href="#for_algebras_over_1operads">For algebras over (∞,1)-operads</a></li></ul></li><li><a href="#reference">Reference</a></li></ul></div> <h2 id="idea">Idea</h2> <p>In <a class='existingWikiWord' href='/nlab/show/higher+algebra'>higher algebra</a>/<a class='existingWikiWord' href='/nlab/show/higher+category+theory'>higher category theory</a> one can define (generalized) <a class='existingWikiWord' href='/nlab/show/algebra'>algebraic structures</a> <a class='existingWikiWord' href='/nlab/show/internalization'>internal</a> to <a class='existingWikiWord' href='/nlab/show/categories'>categories</a> which themselves are equipped with certain algebraic structure, in fact with the <em>same kind</em> of algebraic structure. In (<a href="#BaezDolan">Baez-Dolan 97</a>) this has been called the <em>microcosm principle</em>.</p> <blockquote> <p><strong>Microcosm principle</strong>: <em>Certain <a class='existingWikiWord' href='/nlab/show/algebra'>algebraic structures</a> can be defined <a class='existingWikiWord' href='/nlab/show/internalization'>in</a> any category equipped with a <a class='existingWikiWord' href='/nlab/show/categorification'>categorified version</a> of the same structure.</em></p> </blockquote> <p>A standard example is the notion of <em><a class='existingWikiWord' href='/nlab/show/monoid+object'>monoid object</a></em> which make sense in a <em><a class='existingWikiWord' href='/nlab/show/monoidal+category'>monoidal category</a></em>, and the notion of <em><a class='existingWikiWord' href='/nlab/show/commutative+monoid+object'>commutative monoid object</a></em> which makes sense in a <em><a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+category'>symmetric monoidal category</a></em>.</p> <p>Moreover, in many cases the “internal” version of the structure can be identified with <a class='existingWikiWord' href='/nlab/show/lax+morphisms'>lax morphisms</a> of categorified structures with <a class='existingWikiWord' href='/nlab/show/terminal+object'>terminal</a> domain. For instance, a monoid in a monoidal category is equivalently a <a class='existingWikiWord' href='/nlab/show/lax+monoidal+functor'>lax monoidal functor</a> with terminal domain. More generally, the categorified structure can often be generalized to a “virtual” (<a class='existingWikiWord' href='/nlab/show/generalized+multicategory'>generalized multicategory</a>-like) structure.</p> <p>The term “microcosm principle” arises from the idea that the monoid object (the ‘microcosm’) can only thrive in a category (a ‘macrocosm’) that resembles itself.</p> <h2 id="examples">Examples</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/monoid+objects'>monoid objects</a> can be defined in <a class='existingWikiWord' href='/nlab/show/monoidal+categories'>monoidal categories</a> (categorified monoids), or more generally <a class='existingWikiWord' href='/nlab/show/multicategories'>multicategories</a> (virtual monoidal categories).</li> <li><a class='existingWikiWord' href='/nlab/show/commutative+monoid+objects'>commutative monoid objects</a> can be defined in <a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+categories'>symmetric monoidal categories</a>, or more generally in <a class='existingWikiWord' href='/nlab/show/braided+monoidal+categories'>braided monoidal categories</a>.</li> <li>Categorifying these examples, <a class='existingWikiWord' href='/nlab/show/monoidal+categories'>monoidal categories</a> are instances of <a class='existingWikiWord' href='/nlab/show/pseudomonoids'>pseudomonoids</a>, which can be defined in any <a class='existingWikiWord' href='/nlab/show/monoidal+bicategory'>monoidal bicategory</a>. There are also versions for braided, symmetric, closed, compact closed, and so on.</li> <li><span class='newWikiWord'>traces<a href='/nlab/new/trace+in+a+bicategory'>?</a></span> can be defined in bicategories equipped with a <a class='existingWikiWord' href='/nlab/show/shadow'>shadow</a> (a categorified trace).</li> <li><a class='existingWikiWord' href='/nlab/show/categories'>categories</a> are instances of <a class='existingWikiWord' href='/nlab/show/monads'>monads</a>, which can be defined in any <a class='existingWikiWord' href='/nlab/show/bicategory'>bicategory</a> or more generally any <a class='existingWikiWord' href='/nlab/show/double+category'>double category</a> (both a kind of categorified category) or <a class='existingWikiWord' href='/nlab/show/virtual+double+category'>virtual double category</a>. This generalization includes <a class='existingWikiWord' href='/nlab/show/internal+categories'>internal categories</a> and <a class='existingWikiWord' href='/nlab/show/enriched+categories'>enriched categories</a>.</li> <li><a class='existingWikiWord' href='/nlab/show/enriched+categories'>enriched categories</a> are also objects of the free cocompletion of an <a class='existingWikiWord' href='/nlab/show/enriched+bicategory'>enriched bicategory</a> under a kind of <a class='existingWikiWord' href='/nlab/show/Kleisli+object'>Kleisli object</a>.</li> <li><a class='existingWikiWord' href='/nlab/show/generalized+multicategories'>generalized multicategories</a> are naturally defined internal to a <a class='existingWikiWord' href='/nlab/show/virtual+double+category'>virtual double category</a>, which is itself a kind of generalized multicategory.</li> <li><a class='existingWikiWord' href='/nlab/show/double+categories'>double categories</a> are instances of <span class='newWikiWord'>intermonads<a href='/nlab/new/intermonads'>?</a></span>, which can be defined in any <a class='existingWikiWord' href='/nlab/show/intercategory'>intercategory</a> (a categorified double category).</li> <li><span class='newWikiWord'>duoids<a href='/nlab/new/duoids'>?</a></span> (objects with two monoidal structures) can be defined in any <a class='existingWikiWord' href='/nlab/show/duoidal+category'>duoidal category</a> (a category with two monoidal structures).</li> <li><a class='existingWikiWord' href='/nlab/show/Frobenius+algebras'>Frobenius algebras</a> can be defined in any <a class='existingWikiWord' href='/nlab/show/star-autonomous+category'>star-autonomous category</a>, and cocomplete <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>\ast</annotation></semantics></math>-autonomous posets are Frobenius algebras in the <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>\ast</annotation></semantics></math>-autonomous category <a class='existingWikiWord' href='/nlab/show/Sup'>Sup</a>, while <a class='existingWikiWord' href='/nlab/show/promonoidal+category'>pro</a>-<math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>\ast</annotation></semantics></math>-autonomous categories are <a class='existingWikiWord' href='/nlab/show/Frobenius+pseudomonoids'>Frobenius pseudomonoids</a> in the <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>\ast</annotation></semantics></math>-autonomous (indeed, compact closed) bicategory <a class='existingWikiWord' href='/nlab/show/Prof'>Prof</a>.</li> <li>The prototypical example of a <a class='existingWikiWord' href='/nlab/show/2-fibration'>2-fibration</a> consists of internal <a class='existingWikiWord' href='/nlab/show/fibrations+in+a+2-category'>fibrations in a 2-category</a>.</li> <li><a class='existingWikiWord' href='/nlab/show/type+theories'>type theories</a> can be defined inside a “2-type theory”; see <a class='existingWikiWord' href='/nlab/show/adjoint+type+theory'>adjoint type theory</a> and <a class='existingWikiWord' href='/nlab/show/logical+framework'>logical framework</a>.</li> </ul> <h2 id="formalizations">Formalizations</h2> <p>The microcosm principle is a general heuristic, but in some contexts, general versions of it can be defined and proven formally. These do not generally include all examples, but often include many of them.</p> <h3 id="for_categorical_operad_algebras">For <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-categorical operad algebras</h3> <p><a href="#BaezDolan">Baez and Dolan</a> gave the following precise definition: for any <a class='existingWikiWord' href='/nlab/show/operad'>operad</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math>, an <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-categorical <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math>-algebra internal to an <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n+1)</annotation></semantics></math>-categorical <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math>-algebra <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/lax+morphism'>lax morphism</a> from the <a class='existingWikiWord' href='/nlab/show/terminal+object'>terminal</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n+1)</annotation></semantics></math>-categorical <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math>-algebra to <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'>A.</annotation></semantics></math></p> <h3 id="for_categorical_algebras_over_cartesian_monads">For categorical algebras over cartesian monads</h3> <p><a href="#Batanin">Batanin</a> gave the following definition: for a <a class='existingWikiWord' href='/nlab/show/cartesian+monad'>cartesian monad</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>, an internal <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-algebra in an algebra <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> for the extension of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> to <a class='existingWikiWord' href='/nlab/show/internal+categories'>internal categories</a> is a lax <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-morphism from the <a class='existingWikiWord' href='/nlab/show/terminal+object'>terminal</a> such algebra to <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>. More generally, <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> can be an algebra for some other monad, for instance which builds in some commutativity.</p> <h3 id="for_categorical_algebras_over_lawvere_theories">For categorical algebras over Lawvere theories</h3> <p><a href="#HJS">Hasuo, Jacobs, and Sokolova</a> gave the following very similar definition: for a <a class='existingWikiWord' href='/nlab/show/Lawvere+theory'>Lawvere theory</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>, an internal <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>-algebra in a categorical <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>-algebra (an <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>-algebra in <a class='existingWikiWord' href='/nlab/show/Cat'>Cat</a>) is a lax <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>-morphism from the <a class='existingWikiWord' href='/nlab/show/terminal+object'>terminal</a> such algebra to <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>. They used this to prove a general result about the liftings of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>-algebra structures to coalgebras.</p> <h3 id="for_algebras_over_1operads">For algebras over (∞,1)-operads</h3> <p>Another such formalization was given (independently) in (<a href="##Lurie">Lurie</a>) in the context of <a class='existingWikiWord' href='/nlab/show/higher+algebra'>higher algebra</a> in <a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homotopy theory</a>/<a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-category+theory'>(∞,1)-category theory</a>:</p> <p>given an <a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-operad'>(∞,1)-operad</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math> we have:</p> <ol> <li> <p>an <a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-category'>(∞,1)-category</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> equipped with algebraic structure as encoded by <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math> – an <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category'>monoidal (∞,1)-category</a> – is equivalently encoded by a <a class='existingWikiWord' href='/nlab/show/coCartesian+fibration+of+%28%E2%88%9E%2C1%29-operads'>coCartesian fibration of (∞,1)-operads</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒞</mi> <mo>⊗</mo></msup><mo>→</mo><msup><mi>𝒪</mi> <mo>⊗</mo></msup></mrow><annotation encoding='application/x-tex'>\mathcal{C}^\otimes \to \mathcal{O}^\otimes</annotation></semantics></math>;</p> </li> <li> <p>an <a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-algebra+over+an+%28%E2%88%9E%2C1%29-operad'>(∞,1)-algebra over an (∞,1)-operad</a> over <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math> can be defined internal to such a <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math>-monoidal <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-category <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> and is equivalently given by a <a class='existingWikiWord' href='/nlab/show/section'>section</a> of this map.</p> </li> </ol> <p>See at <em><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-algebra+over+an+%28%E2%88%9E%2C1%29-operad'>(∞,1)-algebra over an (∞,1)-operad</a></em> for examples and further details.</p> <h2 id="reference">Reference</h2> <p>The term “microcosm principle” was coined in</p> <ul id="BaezDolan"> <li><a class='existingWikiWord' href='/nlab/show/John+Baez'>John Baez</a>, <a class='existingWikiWord' href='/nlab/show/James+Dolan'>James Dolan</a>, <em>Higher-Dimensional Algebra III</em> (<a href="http://arxiv.org/abs/q-alg/9702014">arXiv:q-alg/9702014</a>)</li> </ul> <p>Discussion is in</p> <ul> <li><a href="http://golem.ph.utexas.edu/category/2008/12/the_microcosm_principle.html">The Microcosm Principle</a></li> </ul> <p>Other formalizations can be found in</p> <ul id="Batanin"> <li><a class='existingWikiWord' href='/nlab/show/Michael+Batanin'>Michael Batanin</a>, <em>The Eckman-Hilton argument and higher operads</em>, <a href="https://arxiv.org/abs/math/0207281">arxiv:0207281</a></li> </ul> <ul id="HJS"> <li>Ichiro Hasuo, <a class='existingWikiWord' href='/nlab/show/Bart+Jacobs'>Bart Jacobs</a>, and Ana Sokolova. (2008) <em>The Microcosm Principle and Concurrency in Coalgebra</em>. In: Amadio R. (eds) <em>Foundations of Software Science and Computational Structures</em>. FoSSaCS 2008. Lecture Notes in Computer Science, vol 4962. Springer, Berlin, Heidelberg. <a href="https://doi.org/10.1007/978-3-540-78499-9_18">doi:10.1007/978-3-540-78499-9_18</a>, <a href="http://cs.uni-salzburg.at/~anas/papers/HasuoJS.pdf">paper</a>, <a href="http://cs.uni-salzburg.at/~anas/papers/MicrocosmPresentation.pdf">slides</a>, <a href="http://cs.uni-salzburg.at/~anas/papers/Ohrid2008ana.pdf">more slides</a></li> </ul> <ul id="Lurie"> <li> <p>Ichiro Hasuo, Chris Heunen, Bart Jacobs, Ana Sokolova, <em>Coalgebraic Components in a Many-Sorted Microcosm</em>, International Conference on Algebra and Coalgebra in Computer Science CALCO 2009: Algebra and Coalgebra in Computer Science pp 64-80, (<a href="http://homepages.inf.ed.ac.uk/cheunen/publications/2009/component/component.pdf">pdf</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Jacob+Lurie'>Jacob Lurie</a>, <em><a class='existingWikiWord' href='/nlab/show/Higher+Algebra'>Higher Algebra</a></em></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 7, 2018 at 02:31:00. See the <a href="https://ncatlab.org/nlab/history/microcosm principle" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/microcosm+principle" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a> | <a href="/nlab/revision/microcosm+principle/14" accesskey="B" class="navlink" id="to_previous_revision" rel="nofollow">Back in time</a> <span class='revisions'>(14 revisions)</span> | <a href="/nlab/show/diff/microcosm+principle" accesskey="C" class="navlink" id="see_changes" rel="nofollow">See changes</a> | <a href="/nlab/history/microcosm+principle" accesskey="S" class="navlink" id="history" rel="nofollow">History</a> | <a href="https://ncatlab.org/nlab/show/microcosm principle/cite" style="color: black">Cite</a> <span class="views"> | Views: <a href="/nlab/print/microcosm+principle" accesskey="p" id="view_print" rel="nofollow">Print</a> | <a href="/nlab/tex/microcosm+principle" id="view_tex" rel="nofollow">TeX</a> | <a href="/nlab/source/microcosm+principle" id="view_source" rel="nofollow">Source</a> </span> </div> <div id="footer"> <div>This site is running on <a href="http://golem.ph.utexas.edu/instiki/show/HomePage">Instiki 0.19.7(MML+)</a></div> <div>Powered by <a href="http://rubyonrails.com/">Ruby on Rails</a> 2.3.18</div> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>