Categorification (original) (raw)
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'.
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| dbo: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) |
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| 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) |
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