Categorification

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, categorification is the process of replacing <a href="/facts/Set_theory/1ptfsvUP">set-theoretic</a> <a href="/facts/Theorem/f2Pe1FMD">theorems</a> with <a href="/facts/Category_theory/6zS3DXPa">category-theoretic</a> analogues. Categorification, when done successfully, replaces <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> with <a href="/facts/Category_(mathematics)/7WzxUThf">categories</a>, <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> with <a href="/facts/Functor/l2u3rIBE">functors</a>, and <a href="/facts/Equation/pxFzLAVR">equations</a> with <a href="/facts/Natural_transformation/xwPumTHt">natural isomorphisms</a> of functors satisfying additional properties. The term was coined by <a href="/facts/Louis_Crane/5kJHoGqP">Louis Crane</a>.
The reverse of categorification is the process of decategorification. Decategorification is a systematic process by which <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> objects in a category are identified as <a href="/facts/Equality_(mathematics)/AQCuumLg">equal</a>. Whereas decategorification is a straightforward process, categorification is usually much less straightforward. In the <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a> of <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebras</a>, <a href="/facts/Module_over_a_ring/jP0LIhFL">modules</a> 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 '<a href="/facts/Generalization/v0b1pwUx">generalization</a>', and not like '<a href="/facts/Sheafification/V3MnSb2f">sheafification</a>'.

Categorification open-in-new

Categorification