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Study of categorified structures
especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology
Higher-dimensional_algebra
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
General theory of mathematical structures
to the ordinal number ω. Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra, a concept introduced by
Category_theory
In mathematics, invertible homomorphism
isomorphism class of a finite-dimensional vector space can be identified with the non-negative integer representing its dimension. The classification of finite
Isomorphism
Embedding of categories into functor categories
It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. The
Yoneda_lemma
Central object of study in category theory
finite-dimensional vector space is again a finite-dimensional vector space of the same dimension, and these are thus isomorphic, since dimension is the
Natural_transformation
Mathematical construction used in homotopy theory
applications in algebraic geometry where CW complexes do not naturally exist. Simplicial sets can be viewed as a higher-dimensional generalization of
Simplicial_set
English mathematician (1935–2024)
(in reference to Heyting-algebra higher-dimensional-algebra hyperalgebras Łukasiewicz-Moisil-algebras meta-logics MV-algebras on 2007-07-11) Cited in Baez
Ronald_Brown_(mathematician)
Applications of category theory
natural language with compact closed categories and Frobenius algebras", Logic and Algebraic Structures in Quantum Computing, Cambridge University Press
Applied_category_theory
Property of a mathematical space
A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because
Dimension
Hypothesis in mathematical category theory
Definition 1.4.4. (ver.arXiv) Baez, John C.; Dolan, James (1995). "Higher-dimensional algebra and topological quantum field theory". Journal of Mathematical
Homotopy_hypothesis
Characterizing property of mathematical constructions
property is used rather than the concrete details. For example, the tensor algebra of a vector space is slightly complicated to construct, but much easier
Universal_property
algebraic topology studies an aspect of algebraic topology that involves (inevitably noncommutative) higher-dimensional algebras. Many of the higher-dimensional
Nonabelian_algebraic_topology
Category admitting tensor products
over a field K, with the one-dimensional vector space K serving as the unit. K-FdVect (the category of finite-dimensional vector spaces) by extension fits
Monoidal_category
Branch of mathematics
theorem Algebraic K-theory Exact sequence Glossary of algebraic topology Grothendieck topology Higher category theory Higher-dimensional algebra Homological
Algebraic_topology
Method for producing composition algebras
finite-dimensional normed division algebras over the real numbers, while the Frobenius theorem states that the first three are the only finite-dimensional associative
Cayley–Dickson_construction
Concept in category theory
input's structure or properties before mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the
Forgetful_functor
Theorem in category theory
category Quotient category Subcategory Higher category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category
Lawvere's_fixed-point_theorem
Abstract mathematics relationship
However, in contrast to the situation common for isomorphisms in an algebraic setting, the composite of the functor and its "inverse" is not necessarily
Equivalence_of_categories
Map (arrow) between two objects of a category
that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions
Morphism
Space with one dimension
if the algebra is of higher dimensionality. One dimensional coordinate systems include the number line. Number line Univariate Zero-dimensional space Гущин
One-dimensional_space
Special objects used in (mathematical) category theory
and K-Vect, the category of vector spaces over a field. See Zero object (algebra) for details. This is the origin of the term "zero object". In Ring, the
Initial_and_terminal_objects
Mapping between categories
in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects
Functor
Functors which are surjective and injective on hom-sets
(second ed.). Springer. ISBN 0-387-98403-8. Jacobson, Nathan (2009). Basic algebra. Vol. 2 (2nd ed.). Dover. ISBN 978-0-486-47187-7. Riehl, Emily (November
Full_and_faithful_functors
Type of category in category theory
particular, the category of vector spaces over a field K is additive. The algebra of matrices over a ring, thought of as a category as described below, is
Additive_category
Category-theoretic construction
commutative R-algebras is the tensor product. In the category of (noncommutative) R-algebras, the coproduct is a quotient of the tensor algebra (see Free
Coproduct
Construction in category theory
sense of universal algebra, that is, a type of algebraic structures, whose axioms are unconditional (fields do not form an algebra, since zero does not
Inverse_limit
Mathematical concept
Zbl 0906.18001. Borceux, Francis (1994). "Limits". Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52].
Limit_(category_theory)
Concept in mathematical category theory
field k and a group (or a Lie algebra over k), the category of all k-linear representations of the group (or of the Lie algebra) is a symmetric monoidal category
Symmetric_monoidal_category
Category with direct sums and certain types of kernels and cokernels
properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory
Abelian_category
and (in the last case) uniqueness of certain lifts. In algebraic topology and homological algebra, tensor product and the Hom functor are adjoint; however
Lift_(mathematics)
Type of category in category theory
topology, and currying, together with apply, provide the adjoint. A Heyting algebra is a Cartesian closed (bounded) lattice. An important example arises from
Cartesian_closed_category
Set of arguments where two or more functions have the same value
terminology comes from, and why it is most common in the context of abstract algebra: The difference kernel of f and g is simply the kernel of the difference
Equaliser_(mathematics)
Overview of and topical guide to category theory
Functor Natural transformation Homological algebra Diagram chasing Topos theory Enriched category theory Higher category theory Categorical logic Applied
Outline_of_category_theory
Generalized object in category theory
Definition 2.1.1 in Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50–51, 53 [i.e. 52].
Product_(category_theory)
Branch of mathematics
1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth
Linear_algebra
Endofunctor on the category V of finite-dimensional vector spaces
In algebra, a polynomial functor is an endofunctor on the category V {\displaystyle {\mathcal {V}}} of finite-dimensional vector spaces that depends polynomially
Polynomial_functor
Functor type
Thomas. Algebra. Springer-Verlag. p. 470. ISBN 3-540-90518-9. Nourani, Cyrus (19 April 2016). A Functorial Model Theory: Newer Applications to Algebraic Topology
Representable_functor
Most general completion of a commutative square given two morphisms with same codomain
required to be unique. Pullbacks in differential geometry Join (relational algebra) Mitchell, p. 9 Lee, John M. (2003). "Smooth Manifolds". Graduate Texts
Pullback_(category_theory)
Mathematical object that generalizes the standard notions of sets and functions
(revised ed.), MR 2178101. Borceux, Francis (1994), "Handbook of Categorical Algebra", Encyclopedia of Mathematics and its Applications, vol. 50–52, Cambridge:
Category_(mathematics)
Monoidal category
finite-dimensional linear representations of G. More generally, it may be that fiber functors F as above only exists to categories of finite-dimensional vector
Tannakian_formalism
Relationship between two functors abstracting many common constructions
80–82. ISBN 0-387-98403-8. Zbl 0906.18001. Baez, John C. (1996). "Higher-Dimensional Algebra II: 2-Hilbert Spaces". arXiv:q-alg/9609018. Kan, Daniel M. (1958)
Adjoint_functors
Connects set theory with category theory
analogues. Higher category theory Higher-dimensional algebra Categorical ring Crane, Louis; Frenkel, Igor B. (1994-10-01). "Four-dimensional topological
Categorification
Relation of categories in category theory
arises in the Boolean algebras theory: Boolean algebras is isomorphic to the category of Boolean rings. Given a Boolean algebra B, we turn B into a Boolean
Isomorphism_of_categories
Generalization of category
Three-Dimensional Category Theory. pp. 21–34. doi:10.1017/CBO9781139542333.003. ISBN 978-1-139-54233-3. Khan, Adeel A. (2023). "Lectures on algebraic stacks"
2-category
Category whose hom sets have algebraic structure
given field are enriched over themselves, where the morphisms inherit the algebraic structure "pointwise". More generally, preadditive categories are categories
Enriched_category
Theory of algebraic structures in general
algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures
Universal_algebra
Collection of maps which give the same result
commutative diagrams play the role in category theory that equations play in algebra. A commutative diagram often consists of three parts: objects (also known
Commutative_diagram
Gamma matrices for arbitrary Clifford algebras
In mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a
Higher-dimensional gamma matrices
Higher-dimensional_gamma_matrices
Category whose objects and morphisms are inside a bigger category
Jaap van Oosten. "Basic category theory" (PDF). Freyd, Peter (1991). "Algebraically complete categories". Proceedings of the International Conference on
Subcategory
Algebra associated to any vector space
introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude of
Exterior_algebra
Mathematical category
possess a notion of localization. Grothendieck topoi find applications in algebraic geometry. They are generalized by elementary topoi, which are used in
Topos
Category theory constructs
of (what is now known as) a Kan extension from 1956 was in homological algebra to compute derived functors. In Categories for the Working Mathematician
Kan_extension
Graphical representation of a morphism
Frobenius algebra, the cuts are unary operators on homsets that axiomatise logical negation. This makes string diagrams a sound and complete two-dimensional deduction
String_diagram
Most general completion of a commutative square given two morphisms with same domain
algebraic topology. University of Chicago Press, 1999. An introduction to categorical approaches to algebraic topology: the focus is on the algebra,
Pushout_(category_theory)
Variant of the notion of the center of a monoid, group, or ring to a category
Journal of Pure and Applied Algebra, 71 (1): 43–51, doi:10.1016/0022-4049(91)90039-5, MR 1107651. Lurie, Jacob (2017), Higher Algebra Majid, Shahn (1991). "Representations
Center_(category_theory)
The derived category of an abelian category is much used in homological algebra. It is the localization of the category of chain complexes (up to homotopy)
Localization_of_a_category
Categorical generalization of a function space in set theory
( x , y ) . {\displaystyle \lambda g(x)=y\mapsto g(x,y).\,} A Heyting algebra H {\displaystyle H} is just a bounded lattice that has all exponential
Exponential_object
Mathematics construct
"Functorial semantics of algebraic theories" and "Some algebraic problems in the context of functorial semantics of algebraic theories". http://www.tac
Comma_category
Category whose hom objects correspond (di-)naturally to objects in itself
[1966]. "Closed categories". Proceedings of the Conference on Categorical Algebra. (La Jolla, 1965. Springer. pp. 421–562. doi:10.1007/978-3-642-99902-4_22
Closed_category
Surjective homomorphism
abstract algebra and universal algebra define an epimorphism simply as an onto or surjective homomorphism. Every epimorphism in this algebraic sense is
Epimorphism
Algebra based on a vector space with a quadratic form
two-dimensional algebra generated by e1 that squares to −1, and is algebra-isomorphic to C, the field of complex numbers. Cl1,0(R) is a two-dimensional algebra
Clifford_algebra
Mathematical category with weak equivalences, fibrations and cofibrations
homological algebra. Homology can then be viewed as a type of homotopy, allowing generalizations of homology to other objects, such as groups and R-algebras, one
Model_category
Aspect of category theory
category Quotient category Subcategory Higher category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category
Coequalizer
Object in category theory
binary coproducts (denoted by +), an NNO can be defined as the initial algebra of the endofunctor that acts on objects by X ↦ 1 + X and on arrows by f
Natural_numbers_object
Construction in algebra
then the Hopf algebra is said to be involutive (and the underlying algebra with involution is a *-algebra). If H is finite-dimensional semisimple over
Hopf_algebra
n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra. Here, n {\displaystyle
N-group_(category_theory)
Category in which all small limits exist
finite sets The category of finite abelian groups The category of finite-dimensional vector spaces Any (pre)abelian category is finitely complete and finitely
Complete_category
Mathematical concept
category Quotient category Subcategory Higher category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category
End_(category_theory)
Category theory
associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli category is one of two extremal solutions to the question:
Kleisli_category
Category theory concept
commutative rings. This is because the structure of an A {\displaystyle A} -algebra on a commutative ring B {\displaystyle B} is directly encoded by a ring
Overcategory
measure of nonassociativity of Q. In higher-dimensional algebra, where there may be non-identity morphisms between algebraic expressions, an associator is an
Associator
Classification of topological quantum field theories
for the point. Cobordism Baez, John C.; Dolan, James (1995). "Higher-dimensional algebra and topological quantum field theory". Journal of Mathematical
Cobordism_hypothesis
Cohomology theory for Lie algebras
homological algebra. Cambridge University Press. p. 240. Baez, John C.; Crans, Alissa S. (2004). "Higher-dimensional algebra VI: Lie 2-algebras". Theory
Lie_algebra_cohomology
Mathematical category formed by reversing morphisms
equivalent to the category of commutative Von Neumann algebras (with normal unital homomorphisms of *-algebras). Opposite preserves products: ( C × D ) op ≅ C
Opposite_category
Algebra combining both supersymmetry and conformal symmetry
superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group
Superconformal_algebra
Injective homomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with
Monomorphism
Specific algebraic group
commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled
Algebraic_torus
Hypercomplex number system
This equation means that the octonions form a composition algebra. The higher-dimensional algebras defined by the Cayley–Dickson construction (starting with
Octonion
Quotient space of a codomain of a linear map by the map's image
and Q itself is called the cokernel of f. In many situations in abstract algebra, such as for abelian groups, vector spaces or modules, the cokernel of
Cokernel
Correspondence between properties of a category and its opposite
notions. Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called
Dual_(category_theory)
In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more
Fundamental_groupoid
especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension. A double
Double_groupoid
able to explicitly study the structure behind those equalities. Higher-dimensional algebra the study of categorified structures. Hodge theory a method for
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Construction in category theory
ISBN 0-387-98403-8. Borceux, Francis (1994). "Limits". Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52].
Cone_(category_theory)
Special case of colimit in category theory
case of limits in category theory. We will first give the definition for algebraic structures like groups and modules, and then the general definition, which
Direct_limit
Branch of mathematics
as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces. It
Multilinear_algebra
Product of two categories, in category theory
Definition 1.6.5 in Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52].
Product_category
pp. 20–23. ISBN 9780387977102. May, J. P. (1999). A Concise Course in Algebraic Topology (PDF). University of Chicago Press. p. 16. ISBN 0-226-51183-9
Diagonal_functor
Geometric space with four dimensions
tessarines and coquaternions were introduced as other four-dimensional algebras over R. Higher-dimensional non-Euclidean spaces were put on a firm footing by
Four-dimensional_space
Generalization of a Lie algebra
"Higher-Dimensional Algebra VI: Lie 2-Algebras". Theory and Applications of Categories. 12 (15): 492–528. https://ncatlab.org/nlab/show/Lie+2-algebra https://golem
Lie_n-algebra
248-dimensional exceptional simple Lie group
Lie algebra Ek for every integer k ≥ 3. The largest value of k for which Ek is finite-dimensional is k = 8, that is, Ek is infinite-dimensional for any
E8_(mathematics)
A. (2003). "Higher-dimensional algebra V: 2-groups". arXiv:math.QA/0307200. Brown, R. (1999). "Groupoids and crossed objects in algebraic topology" (PDF)
Crossed_module
Concept in mathematics
Homotopy Theory. A.M.S. p. 253. ISBN 0-8218-3922-5. Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9
Tensor–hom_adjunction
Algebra describing 2D conformal symmetry
Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal
Virasoro_algebra
Baez, John C.; Dolan, James (10 May 1998). "Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes". Advances in Mathematics. 135 (2):
3-category
Mathematical category whose hom sets form Abelian groups
particular: the category of vector spaces over a field K {\displaystyle K} . The algebra of matrices over a ring, thought of as a category as described in the article
Preadditive_category
category Quotient category Subcategory Higher category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category
Essentially surjective functor
Essentially_surjective_functor
Bi-universal property in category theory
category Quotient category Subcategory Higher category theory Key concepts Enriched category Higher-dimensional algebra Homotopy hypothesis Model category
Zero_morphism
Branch of biology
Glazebrook JF (2006). "Complex Non-linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic
Mathematical and theoretical biology
Mathematical_and_theoretical_biology
HIGHER DIMENSIONAL-ALGEBRA
HIGHER DIMENSIONAL-ALGEBRA
Biblical
a digger
Male
English
Variant spelling of English Hugh, HUGHE means "heart," "mind," or "spirit."
Girl/Female
Muslim
Higher, Highest
Girl/Female
Indian
Higher, Highest
Girl/Female
Hindu
Three dimensional
Surname or Lastname
English (mainly Sussex and Kent)
English (mainly Sussex and Kent) : topographic name from Middle English hilder ‘dweller on a slope’ (from Old English hylde ‘slope’).
Surname or Lastname
Scottish
Scottish : variant spelling of Biggar.English : occupational name for a builder, from Middle English bigger ‘(house) builder’, an agent derivative of bigge(n) ‘to build’ (from Old Norse byggja).
Surname or Lastname
English
English : variant of Haggard.English : variant of Hager.
Boy/Male
Biblical
A digger.
Boy/Male
Hindu, Indian
Dimensions
Girl/Female
Indian, Telugu
Uni-dimensional
Boy/Male
Tamil
Dimensions
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Three Dimentional
Girl/Female
Hindu, Indian
Three Dimension
Girl/Female
Tamil
Trikaya | தà¯à®°à®¿à®•à®¾à®¯à®¾
Three dimensional
Trikaya | தà¯à®°à®¿à®•à®¾à®¯à®¾
Surname or Lastname
English
English : variant of Highley.
Male
Swedish
Swedish form of Old Norse Dagr, DAGHER means "day."
Boy/Male
Tamil
Trigun | தà¯à®°à®¿à®•à¯à®£
The three dimensions
Trigun | தà¯à®°à®¿à®•à¯à®£
Girl/Female
Gujarati, Indian, Kannada
Dimension; Purity
Girl/Female
Tamil
Triguni | தà¯à®°à¯€à®•à¯‚நீ
The three dimensions
HIGHER DIMENSIONAL-ALGEBRA
HIGHER DIMENSIONAL-ALGEBRA
Boy/Male
American, Australian, Finnish, French, German, Latin, Portuguese, Spanish
Moorish; Dark-skinned; A Moor; Dark Skinned
Boy/Male
Sikh
Goddess Laxmi
Girl/Female
English Teutonic
Shining battlemaid.
Girl/Female
Hindu, Indian
Rich; Goddess Lakshmi
Girl/Female
Muslim
Pure Love
Boy/Male
Christian, Finnish, French, Hebrew, Hindu, Indian
Gift of God; Diminutive of Matthew; Gift of the Lord
Surname or Lastname
English (East Anglia)
English (East Anglia) : from a short form of the personal name Vincent.Hungarian : variant of Vincze.
Boy/Male
Tamil
Sri Kanth | à®·à¯à®°à¯€ கஂட Â
Sri Hari, Beloved of Sri
Surname or Lastname
English
English : occupational name for the servant (Middle English man) of a nobleman (Middle English hold(e)).English : variant of Oldman, derived from Old English (e)ald ‘old’ + mann ‘man’.North German (Holdmann) : topographic name from Middle Low German holt ‘small wood’ + man ‘man’.
Girl/Female
Indian
Princess; Moon; Priest
HIGHER DIMENSIONAL-ALGEBRA
HIGHER DIMENSIONAL-ALGEBRA
HIGHER DIMENSIONAL-ALGEBRA
HIGHER DIMENSIONAL-ALGEBRA
HIGHER DIMENSIONAL-ALGEBRA
n.
The degree of manifoldness of a quantity; as, time is quantity having one dimension; volume has three dimensions, relative to extension.
conj. Either
precedes two, or more, coordinate words or phrases, and is introductory to an alternative. It is correlative to or.
v. t.
To convey by a lighter, as to or from the shore; as, to lighter the cargo of a ship.
a.
Being on the side next or toward the person speaking; nearer; -- correlate of thither and farther; as, on the hither side of a hill.
adv.
To that place; -- opposed to hither.
n.
Measure in a single line, as length, breadth, height, thickness, or circumference; extension; measurement; -- usually, in the plural, measure in length and breadth, or in length, breadth, and thickness; extent; size; as, the dimensions of a room, or of a ship; the dimensions of a farm, of a kingdom.
n.
Extent; reach; scope; importance; as, a project of large dimensions.
a.
Pertaining to dimension.
adv.
In a high manner, or to a high degree; very much; as, highly esteemed.
a.
Being on the farther side from the person speaking; farther; -- a correlative of hither; as, on the thither side of the water.
v. i.
Alt. of Zighyr
v. t.
To whirl hither and thither.
a.
Applied to time: On the thither side of, older than; of more years than. See Hither, a.
a.
Having dimensions.
p. p.
of Hight
a.
Applied to time: On the hither side of, younger than; of fewer years than.
n.
A roving hither and thither.
imp.
of Hight
adv.
To this place; -- used with verbs signifying motion, and implying motion toward the speaker; correlate of hence and thither; as, to come or bring hither.
n.
A literal factor, as numbered in characterizing a term. The term dimensions forms with the cardinal numbers a phrase equivalent to degree with the ordinal; thus, a2b2c is a term of five dimensions, or of the fifth degree.