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Connects set theory with category theory
In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully
Categorification
Generalization of category theory
Mathematics portal Higher-dimensional algebra General abstract nonsense Categorification Coherency (homotopy theory) Lurie, Jacob. Higher Topos Theory (PDF)
Higher_category_theory
Russian mathematician
Khovanov homology for links, which was one of the first examples of categorification. Khovanov graduated from Moscow State School 57 mathematical class
Mikhail_Khovanov
Invariant of mathematical knots
arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail
Khovanov_homology
Italian cyclist (born 1987)
Derived Symplectic Structures in Generalized Donaldson–Thomas Theory and Categorification. In September 2018, she set a new UCI Women's hour record, riding 48
Vittoria_Bussi
Category admitting tensor products
examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the
Monoidal_category
Function in algebraic graph theory
04.003, S2CID 53304001 Helme-Guizon, Laure; Rong, Yongwu (2005), "A categorification of the chromatic polynomial", Algebraic & Geometric Topology, 5 (4):
Chromatic_polynomial
General theory of mathematical structures
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Category_theory
Analysis of datasets using techniques from topology
loosen the stricter restriction of the function. Please refer to the Categorification and cosheaves and Impact on mathematics sections for more information
Topological_data_analysis
Concept in mathematics
} that universally characterizes the tensor-hom adjunction, as the categorification of the remarkably basic law of exponents Z Y X = ( Z X ) Y . {\displaystyle
Tensor–hom_adjunction
In mathematics, invertible homomorphism
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Isomorphism
Intermediate structure between functors and monads
functors with tensorial strength. This may not be the most obvious categorification of the standard definition of applicative functor given below, but
Applicative_functor
Hypothesis in mathematical category theory
1007/BFb0026978. ISBN 978-3-540-63455-3. Baez, John C.; Dolan, James (1998). "Categorification". arXiv:math/9802029. Baez, John C. (2007). "The Homotopy Hypothesis"
Homotopy_hypothesis
Mapping between categories
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Functor
Variant of the notion of the center of a monoid, group, or ring to a category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Center_(category_theory)
Theorem in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Lawvere's_fixed-point_theorem
Central object of study in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Natural_transformation
Mathematical object that generalizes the standard notions of sets and functions
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Category_(mathematics)
Most general completion of a commutative square given two morphisms with same codomain
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Pullback_(category_theory)
Functor that preserves short exact sequences
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Exact_functor
Map (arrow) between two objects of a category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Morphism
Embedding of categories into functor categories
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Yoneda_lemma
Swiss mathematician
He has applied triangulated Calabi–Yau categories to the (additive) categorification of cluster algebras. In 2013, he received an honorary degree from the
Bernhard_Keller
British philosopher
(2005). "Doing Mathematics". Philosophia Mathematica. 13: 106–111. "Categorification as a Heuristic Device", in D. Gillies and C. Cellucci (eds.), Mathematical
David_Corfield
Mathematical category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Topos
Mathematical concept
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
End_(category_theory)
Generalization of a category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Quasi-category
Characterizing property of mathematical constructions
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Universal_property
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Localization_of_a_category
Framework of superstring theory
1016/0550-3213(94)00559-W. S2CID 13889163. Khovanov, Mikhail (2000). "A categorification of the Jones polynomial". Duke Mathematical Journal. 1011 (3): 359–426
M-theory
Mathematical concept
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Limit_(category_theory)
Ukrainian-Swedish mathematician
"Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module". Transactions of the American Mathematical
Volodymyr_Mazorchuk
Set of arguments where two or more functions have the same value
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Equaliser_(mathematics)
Injective homomorphism
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Monomorphism
Functor type
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Representable_functor
Special objects used in (mathematical) category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Initial_and_terminal_objects
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Essentially surjective functor
Essentially_surjective_functor
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Conservative_functor
Relationship between two functors abstracting many common constructions
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Adjoint_functors
Applications of category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Applied_category_theory
Homological construction in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Derived_functor
category theory. categorical probability categorical probability categorification categorification is a process of replacing sets and set-theoretic concepts
Glossary_of_category_theory
Type of category in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Additive_category
In mathematics, collection of classes
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Conglomerate_(mathematics)
Functors which are surjective and injective on hom-sets
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Full_and_faithful_functors
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
N-monoid
Type of category in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Cartesian_closed_category
Generalized object in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Product_(category_theory)
Most general completion of a commutative square given two morphisms with same domain
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Pushout_(category_theory)
Category where each homset contains at most one morphism
cocomplete *-autonomous categories can be considered the respective categorifications of posets, distributive lattices, Heyting algebras, and Boolean algebras
Thin_category
Concept in mathematical category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Symmetric_monoidal_category
Category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Kleisli_category
Category-theoretic construction
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Coproduct
Concept in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Monoidal_functor
Relation of categories in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Isomorphism_of_categories
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Tetracategory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Fundamental_groupoid
Category theory constructs
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Kan_extension
Category with direct sums and certain types of kernels and cokernels
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Abelian_category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Stable_∞-category
Collection of maps which give the same result
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Commutative_diagram
Category theory concept
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Overcategory
Mathematical construction used in homotopy theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Simplicial_set
Construction in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Inverse_limit
Category whose objects and morphisms are inside a bigger category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Subcategory
geometric representation theory, noncommutative algebra, and the theory of categorification." In 2010 in Hyderabad he was an invited speaker with talk, Finite
Ivan_Losev_(mathematician)
Monoidal category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Tannakian_formalism
Quotient space of a codomain of a linear map by the map's image
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Cokernel
Abstract mathematics relationship
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Equivalence_of_categories
Aspect of category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Coequalizer
Construction in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Cone_(category_theory)
Mathematical category formed by reversing morphisms
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Opposite_category
Higher category theory concept
doi:10.1006/aima.1997.1695. Baez, John C.; Dolan, James (1998). "Categorification". arXiv:math/9802029. Baez, John C.; Shulman, Michael (2010). "Lectures
Weak_n-category
Indexed collection of objects and morphisms in a category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Diagram_(category_theory)
Endofunctor on the category V of finite-dimensional vector spaces
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Polynomial_functor
Correspondence between properties of a category and its opposite
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Dual_(category_theory)
Mathematical category whose hom sets form Abelian groups
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Preadditive_category
Object in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Natural_numbers_object
Bi-universal property in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Zero_morphism
Category whose hom sets have algebraic structure
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Enriched_category
Generalization of category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
2-category
Technique for proving sets have equal size
counting (proof technique) Combinatorial principles Combinatorial proof Categorification Loehr, Nicholas A. (2011). Bijective Combinatorics. CRC Press. ISBN 143984884X
Bijective_proof
Categorical generalization of a function space in set theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Exponential_object
generalization of a ring eventually leads to the notion of an En-ring. Categorification Higher-dimensional algebra Lie n-algebra John Baez, 2-Rigs in Topology
2-ring
Category whose hom objects correspond (di-)naturally to objects in itself
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Closed_category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Diagonal_functor
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
N-group_(category_theory)
Concept in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Fibred_category
Special case of colimit in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Direct_limit
Category
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Pre-abelian_category
Concept in category theory
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Forgetful_functor
Symmetric monoidal category with a special involution
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Dagger symmetric monoidal category
Dagger_symmetric_monoidal_category
Mathematical category with weak equivalences, fibrations and cofibrations
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Model_category
American mathematician (1960–2024)
the basis for breakthrough research by others on canonical bases , categorification, and geometric representation theory. Letzter was recognized as a fellow
Gail_Letzter
Russian physicist
Known for quantum topology, string theory, special holonomy manifolds, categorification of quantum group invariants, exact solutions of strongly coupled theories
Sergei_Gukov
Mathematics construct
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Comma_category
Algebraic structure with "nice" duality properties
algebras and (1+1)-dimensional TQFTs can be used to explain Khovanov's categorification of the Jones polynomial. Let B be a subring sharing the identity element
Frobenius_algebra
Category in which all small limits exist
Quotient category Subcategory Higher category theory Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model
Complete_category
Categorical procedure
tensors. In this context, skeletonization is the opposite process of categorification, which takes set-theoretic information and turns it into category-theoretic
Skeletonization of fusion categories
Skeletonization_of_fusion_categories
Predicted field theory in physics
1016/j.aim.2012.09.027. S2CID 115176676. Khovanov, Mikhail (2000). "A categorification of the Jones polynomial". Duke Mathematical Journal. 101 (3): 359–426
6D (2,0) superconformal field theory
6D_(2,0)_superconformal_field_theory
CATEGORIFICATION
CATEGORIFICATION
CATEGORIFICATION
CATEGORIFICATION
Surname or Lastname
Irish
Irish : reduced Anglicized form of Gaelic Ó Teimhin ‘descendant of Teimhean’, from teimhean ‘dark’, an adjective from teimhe ‘dusk’, ‘darkness’.English : probably a habitational name for someone from Tyneside in northeast England.
Girl/Female
American, Australian, British, Celtic, Chinese, Christian, Danish, English, French, German, Hebrew, Irish, Latin
Dark Skinned; Great; Dark; Sea of Bitterness; Star of the Sea; Beloved
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : topographic name for someone who lived near an ash tree or ash wood, from Old French fraisne, fresne ‘ash’ (Latin fraxinus).French : habitational name from a place in Vosges named Frain.
Boy/Male
Indian
Boy/Male
Indian
Name of a companion of the prophet
Male
Chinese
thunder.
Girl/Female
Tamil
Lucky
Girl/Female
Arabic, Gujarati, Indian
The World
Boy/Male
Indian
Radiating the beauteous light, Matchless light, Flame
Female
English
Feminine form of English unisex Jocelyn, JOSSLYN means "Gaut."
CATEGORIFICATION
CATEGORIFICATION
CATEGORIFICATION
CATEGORIFICATION
CATEGORIFICATION