categorification

English

Etymology

category +‎ -fication.

Noun

categorification (countable and uncountable, plural categorifications)

1. (category theory) A procedure that defines theorems in terms of category theory by mapping concepts from set theory to category theory.
   • 2008, E. Krenkel, Ramifications of the Geometric Langlands Program, Michael Cowling, Edward Frenkel, Masaki Kashiwara, Alain Valette, David A. Vogan, Jr., Nolan R. Wallach (editors), Representation Theory and Complex Analysis: Lectures Given at the C.I.M.E. Summer School, Springer, Lecture Notes in Mathematics 1931, page 83,

     In Section 3.4 we have already discussed the question of categorification of the algebra of functions on a homogeneous space like $G(F)/K$.

   • 2009, Volodymyr Mazorchuk, Lectures on ${\mathfrak {sl}}_{2}(\mathbb {C} )$-Modules, Imperial College Press, page 221,

     Show that $\Phi \oplus \Phi$ is also a (naïve) homomorphism of naïve categorifications.

   • 2010, Kishore Marathe, Topics in Physical Mathematics, Springer, page 351:

     Khovanov's categorification of the Jones polynomial by Khovanov homology is the subject of Section 11.6.

   • 2011, Robert Wisbauer, Categorical aspects of Hopf algebras, Matilde Marcolli, Deepak Parashar (editors), Quantum Groups and Noncommutative Spaces: Perspectives on Quantum Geometry, Springer Science+Business (Vieweg+Teubner), page 146,

     Since Lawvere's categorification of general algebra, algebras and coalgebras are used as basic notions in universal algebra, logic, and theoretical computer science, for example (e.g. [AdPo], [Gu], [TuPl]).

Derived terms

• decategorification
• horizontal categorification
• vertical categorification

Further reading

• decategorification on nLab
• horizontal categorification on nLab
• vertical categorification on nLab