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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
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
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
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
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
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
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
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
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
In mathematics, invertible homomorphism
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Isomorphism
General theory of mathematical structures
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Category_theory
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
Generalization of a category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Quasi-category
Mathematical object that generalizes the standard notions of sets and functions
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Category_(mathematics)
Variant of the notion of the center of a monoid, group, or ring to a category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Center_(category_theory)
Mathematical concept
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
End_(category_theory)
Map (arrow) between two objects of a category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Morphism
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
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Topos
Mapping between categories
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Functor
Functor that preserves short exact sequences
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Exact_functor
Embedding of categories into functor categories
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Yoneda_lemma
Theorem in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Lawvere's_fixed-point_theorem
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
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
Special objects used in (mathematical) category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Initial_and_terminal_objects
Central object of study in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Natural_transformation
Type of category in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Additive_category
Relationship between two functors abstracting many common constructions
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Adjoint_functors
Applications of category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Applied_category_theory
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
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Limit_(category_theory)
Category with direct sums and certain types of kernels and cokernels
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Abelian_category
Functor type
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Representable_functor
Characterizing property of mathematical constructions
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Universal_property
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Essentially surjective functor
Essentially_surjective_functor
Functors which are surjective and injective on hom-sets
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Full_and_faithful_functors
Category whose hom sets have algebraic structure
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Enriched_category
Quotient space of a codomain of a linear map by the map's image
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Cokernel
Mathematical construction used in homotopy theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Simplicial_set
Most general completion of a commutative square given two morphisms with same domain
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Pushout_(category_theory)
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
Type of category in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Cartesian_closed_category
Generalized object in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Product_(category_theory)
Homological construction in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Derived_functor
Category theory constructs
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Kan_extension
Category-theoretic construction
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Coproduct
Collection of maps which give the same result
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Commutative_diagram
Category whose objects and morphisms are inside a bigger category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Subcategory
Injective homomorphism
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Monomorphism
Set of arguments where two or more functions have the same value
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Equaliser_(mathematics)
Construction in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Inverse_limit
Mathematical category formed by reversing morphisms
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Opposite_category
Most general completion of a commutative square given two morphisms with same codomain
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Pullback_(category_theory)
Mathematical category with weak equivalences, fibrations and cofibrations
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Model_category
In mathematics, collection of classes
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Conglomerate_(mathematics)
Concept in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Forgetful_functor
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Localization_of_a_category
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
Abstract mathematics relationship
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Equivalence_of_categories
Relation of categories in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Isomorphism_of_categories
Category theory concept
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Overcategory
Aspect of category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Coequalizer
category theory. categorical probability categorical probability categorification categorification is a process of replacing sets and set-theoretic concepts
Glossary_of_category_theory
Construction in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Cone_(category_theory)
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
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Diagonal_functor
Concept in mathematical category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Symmetric_monoidal_category
Object in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Natural_numbers_object
Special case of colimit in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Direct_limit
Monoidal category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Tannakian_formalism
Generalization of category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
2-category
Concept in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Monoidal_functor
Bi-universal property in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Zero_morphism
Theory in differential topology
inequalities). The existence of Morse homology "explains", in the sense of categorification, the Morse inequalities. Edward Witten came up with a related construction
Morse_homology
rigidity, which asserts that every fusion ring has finitely many categorifications. A particularly relevant example to condensed matter physics is the
Rank-finiteness
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Refinement_(category_theory)
Indexed collection of objects and morphisms in a category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Diagram_(category_theory)
Category theory
are (equivalence classes of) spans of G-equivariant maps. It is a categorification of the Burnside ring of G. Let G be a finite group (in fact everything
Burnside_category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Fundamental_groupoid
Concept in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Fibred_category
Correspondence between properties of a category and its opposite
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Dual_(category_theory)
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)
Concept in category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Point-surjective_morphism
Category theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Kleisli_category
Endofunctor on the category V of finite-dimensional vector spaces
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Polynomial_functor
Categorical generalization of a function space in set theory
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Exponential_object
Mathematical category whose hom sets form Abelian groups
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Preadditive_category
Category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Pre-abelian_category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Stable_∞-category
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Smooth_functor
Category whose hom objects correspond (di-)naturally to objects in itself
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Closed_category
Mathematics construct
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Comma_category
Category in which all small limits exist
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Complete_category
Category of non-empty finite ordinals and order-preserving maps
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Simplex_category
Russian physicist
Known for quantum topology, string theory, special holonomy manifolds, categorification of quantum group invariants, exact solutions of strongly coupled theories
Sergei_Gukov
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
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
N-group_(category_theory)
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
Lift_(mathematics)
v t e Category theory Key concepts Key concepts Categorification Category Abelian Additive CCC Complete Concrete Forgetful functor Elementary topos Grothendieck
N-monoid
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Boy/Male
Hindu
One of the kauravas
Girl/Female
Indian, Telugu
Decent
Girl/Female
Indian, Telugu
Joy; Praiseworthy
Male
Arthurian
, defender of man.
Boy/Male
Sikh
Skirt of the victor, Victory over suppression
Girl/Female
American, British, English, Greek
Glowing; Modern Variant of Candace; Ancient Hereditary Title Used by Ethiopian Queens; Fire White
Girl/Female
Gujarati, Hindu, Indian
Miracle
Boy/Male
Muslim/Islamic
Beams Rays
Girl/Female
Muslim
Poetess.
Boy/Male
American, British, English
Lives at the Elder Tree
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