About: Categorification

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In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane. Categorification and decategorification are not precise mathematical procedures, but rather a class of possible analogues. They are used in a similar way to the words like 'generalization', and not like 'sheafification'.

Attributes	Values
rdf:type	Election
rdfs:label	Categorification (en) Categorificación (es)
rdfs:comment	In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane. Categorification and decategorification are not precise mathematical procedures, but rather a class of possible analogues. They are used in a similar way to the words like 'generalization', and not like 'sheafification'. (en) En matemáticas, categorificar es el proceso de reemplazar teoremas de la teoría de conjuntos por teoremas equivalentes en teoría de categorías. Para hacer una categorificación exitosa, se deben reemplazar conjuntos con categorías, funciones con , y ecuaciones con isomorfismos naturales de funtores satisfaciendo propiedades adicionales. El término categorificación (en inglés categorification) fue acuñado por Louis Crane. Categorificar y descategorificar no son procedimientos matemáticos precisos, sino una clase de posibles análogos. (es)
dcterms:subject	Algebraic topology Category theory
Wikipage page ID	1596969 (xsd:integer)
Wikipage revision ID	1100529948 (xsd:integer)
Link from a Wikipage to another Wikipage	Module (mathematics) Representation theory Ring of symmetric functions Homology theory Betti number Character (mathematics) Mathematics Generalization Natural transformation Emmy Noether Equality (mathematics) Equation Function (mathematics) Module over a ring Morphism Coproduct Theorem Symmetric functions Lie algebra Littlewood–Richardson rule Cahiers de Topologie et Géométrie Différentielle Catégoriques Combinatorial proof Functor Schur class Category (category theory) Topology Categorical ring Algebraic topology Finite group theory Number theory Khovanov homology Knot theory Product (category theory) Rank of a group Ring (mathematics) Higher-dimensional algebra Category theory Abelian category Higher category theory Free abelian group Grothendieck group Knot invariant Natural number Category theory Set (mathematics) Set theory Symmetric group Specht module Category of finite sets Louis Crane Sheafification
Link from a Wikipage to an external page	http://www.numdam.org/item/CTGDC_1998__39_1_3_0 https://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html.
sameAs	Categorification Categorification Categorification Categorification
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has abstract	In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane. The reverse of categorification is the process of decategorification. Decategorification is a systematic process by which isomorphic objects in a category are identified as equal. Whereas decategorification is a straightforward process, categorification is usually much less straightforward. In the representation theory of Lie algebras, modules over specific algebras are the principal objects of study, and there are several frameworks for what a categorification of such a module should be, e.g., so called (weak) abelian categorifications. Categorification and decategorification are not precise mathematical procedures, but rather a class of possible analogues. They are used in a similar way to the words like 'generalization', and not like 'sheafification'. (en) En matemáticas, categorificar es el proceso de reemplazar teoremas de la teoría de conjuntos por teoremas equivalentes en teoría de categorías. Para hacer una categorificación exitosa, se deben reemplazar conjuntos con categorías, funciones con , y ecuaciones con isomorfismos naturales de funtores satisfaciendo propiedades adicionales. El término categorificación (en inglés categorification) fue acuñado por Louis Crane. El proceso inverso de categorificación es el proceso de descategorificación. Descategorificar es un proceso sistemático en el que objetos isomórfos en una categoría se identifican como iguales. Sistemáticamente, descategorificar es un proceso directo, mientras que categorificar suele ser mucho menos directo. Por ejemplo, en la teoría de representación de álgebras de Lie, los módulos sobre álgebras concretas son los objetos principales de estudio, y existen varias maneras en las que se pueden categorificar dichos módulos; tal es el caso de la llamada categorificación abeliana (débil). Categorificar y descategorificar no son procedimientos matemáticos precisos, sino una clase de posibles análogos. (es)
gold:hypernym	Process
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foaf:isPrimaryTopicOf	wikipedia-en:Categorification
is rdfs:seeAlso of	Gluing axiom
is Link from a Wikipage to another Wikipage of	Quantum logic Monoidal category Posetal category Vittoria Bussi David Corfield Day convolution R-algebroid Timeline of category theory and related mathematics Chromatic polynomial Glossary of areas of mathematics Glossary of category theory Mikhail Khovanov Combinatorial proof H-object K-theory of a category Floer homology Khovanov homology 2-ring Higher-dimensional algebra Ivan Losev (mathematician)

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