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In category theory, a field of mathematics, a category algebra is an associative algebra, defined for any locally finite category and commutative ring
Category_algebra
Operation in algebra and mathematics
this is clear since a monad is a monoid in a certain category, A monad as a tool for studying algebraic gadgets; for example, a group can be described by
Monad_(category_theory)
Directed graph which is also a multigraph
over K is Morita equivalent to the path algebra of its Ext quiver (i.e., they have equivalent module categories). A representation of a quiver Γ is an
Quiver_(mathematics)
Set with operations obeying given axioms
general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes other mathematical structures
Algebraic_structure
Ring that is also a vector space or a module
all R-algebras together with algebra homomorphisms between them form a category, sometimes denoted R-Alg. The subcategory of commutative R-algebras can
Associative_algebra
Branch of mathematics
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations
Abstract_algebra
Branch of mathematics
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems
Algebra
Class of algebraic structures
of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories. A
Variety_(universal_algebra)
Branch of algebra that studies commutative rings
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both
Commutative_algebra
Category whose objects are R-modules and whose morphisms are module homomorphisms
In algebra, given a ring R {\displaystyle R} , the category of left modules over R {\displaystyle R} is the category whose objects are all left modules
Category_of_modules
Mathematical category with weak equivalences, fibrations and cofibrations
recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches
Model_category
Category whose objects are rings and whose morphisms are ring homomorphisms
(= commutative ring) R on an object (= ring) A of Ring is an R-algebra. The category of rings has a number of important subcategories. These include
Category_of_rings
General theory of mathematical structures
Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. In particular
Category_theory
Branch of mathematics
of category theory. A central concept is that of chain complexes, which can be studied through their homology and cohomology. Homological algebra affords
Homological_algebra
Algebraic structure with a binary operation
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with
Magma_(algebra)
Construction in algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative)
Hopf_algebra
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
Universal construction in multilinear algebra
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any order) with multiplication being
Tensor_algebra
Study of categorified structures
especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and
Higher-dimensional_algebra
Category with direct sums and certain types of kernels and cokernels
from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the
Abelian_category
Algebraic structure with addition and multiplication
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted
Ring_(mathematics)
Mathematical structure in abstract algebra
mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of
*-algebra
Mapping between categories
specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such
Functor
Branch of mathematics that studies abstract algebraic structures
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of
Representation_theory
Generalization of category theory
equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces
Higher_category_theory
Representation of a Lie algebra as a set of linear transformations
algebra plays an important role. The universality of this ring says that the category of representations of a Lie algebra is the same as the category
Lie_algebra_representation
Algebraic structure used in analysis
In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket
Lie_algebra
Right inverse of a morphism
{\displaystyle X} . In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms. In an abelian category, if f : X →
Section_(category_theory)
Algebraic structure
Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey
Complete_Heyting_algebra
Study of Hopf algebras via their actions on vector spaces and associative algebras
abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. The representation theory of Hopf algebras is
Representation theory of Hopf algebras
Representation_theory_of_Hopf_algebras
In mathematics, an algebra such as ( R , + , ⋅ ) {\displaystyle (\mathbb {R} ,+,\cdot )} has multiplication ⋅ {\displaystyle \cdot } whose associativity
Homotopy_associative_algebra
History of maths
representation theory and universal algebra; Homological algebra; Homotopical algebra; Topology using categories, including algebraic topology, categorical topology
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Index of articles associated with the same name
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements
Center_(algebra)
Branch of mathematics
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
Mathematical concept in category theory
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) ( M , μ , η ) {\displaystyle (M,\mu ,\eta )} in
Monoid_(category_theory)
Type of monoidal category
quantum field theory, conformal field theory, and quantum algebra. Modular tensor categories were introduced in 1989 by the physicists Greg Moore and Nathan
Modular_tensor_category
"Smallest" commutative algebra that contains a vector space
mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and
Symmetric_algebra
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until
History_of_algebra
Algebraic structure of set algebra
a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used
Σ-algebra
Collection of objects and morphisms
be seen as a small category. Any ordinal number can be seen as a category when viewed as an ordered set. Any monoid (any algebraic structure with a single
Category_(mathematics)
Set of finitely supported functions from a group to a ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free
Group_ring
Free object in the category of associative algebras
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since
Free_algebra
Branch of mathematics that studies algebraic structures
algebra in Wiktionary, the free dictionary. In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures
List of abstract algebra topics
List_of_abstract_algebra_topics
Algebraic manipulation of "true" and "false"
mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables
Boolean_algebra
representation theory of semisimple Lie algebras, Category O (or category O {\displaystyle {\mathcal {O}}} ) is a category whose objects are certain representations
Category_O
Group of mathematical theorems
modules, Lie algebras, and other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences
Isomorphism_theorems
Generalization of vector spaces from fields to rings
central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. In a vector space, the
Module_(mathematics)
Overview of and topical guide to algebraic structures
types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures
Outline of algebraic structures
Outline_of_algebraic_structures
Map (arrow) between two objects of a category
morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from
Morphism
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
Category of mathematics papers in ArXiv
the top-level mathematics categories used by the arXiv, and is a unification of algebraic deformations, Hopf algebras, category theory, topology, noncommutative
Quantum_algebra
Algebraic structure with only one element
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton
Zero_object_(algebra)
Generalization of vector bundles
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric
Coherent_sheaf
Category where each homset contains at most one morphism
initial boolean algebra 2 = { 0 , 1 } {\displaystyle 2=\{0,1\}} regarded as a cartesian monoidal category. All diagrams commute in a thin category. When the
Thin_category
Every Boolean algebra is isomorphic to a certain field of sets
mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem
Stone's representation theorem for Boolean algebras
Stone's_representation_theorem_for_Boolean_algebras
Branch of mathematics
Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b
Linear_algebra
Structure preserving map derived canonically from another map
Rham cohomology are algebraic structures that are functorial, meaning that their definition provides a functor from (e.g.) the category of topological spaces
Induced_homomorphism
Algebra associated to any vector space
In mathematics, the exterior algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle
Exterior_algebra
Function type in category theory
mathematics, specifically in category theory, F-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms
F-algebra
Algebraic structure with "nice" duality properties
theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice
Frobenius_algebra
Freely generated algebraic structure over a given signature
and anarchic algebra. From a category theory perspective, a term algebra is the initial object for the category of all X-generated algebras of the same
Term_algebra
Algebra with braided rigid category of representations
of a braiding on the rigid category of representations of H {\displaystyle H} , which makes quasitriangular Hopf algebras useful in knot theory. The most
Quasitriangular_Hopf_algebra
Overview of and topical guide to category theory
Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. In particular
Outline_of_category_theory
Algebraic structure used in logic
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with
Heyting_algebra
Mathematical structure
initial F {\displaystyle F} -algebras into sums of 'constructors'. Let P be the power set construction on the category of sets, considered as a covariant
F-coalgebra
Branch of mathematics
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants
Algebraic_topology
Generalization of a category
Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200. Groth, Moritz, A short course on infinity-categories (PDF)
Quasi-category
Mathematical object
Frobenius algebra. The stable category of a Frobenius category is canonically a triangulated category. Dagger compact category Tannakian category Theorem
Frobenius_category
Correspondence in universal algebra
Lawvere theories. These two notions are the two main category theoretic formulations of universal algebra. Although these results are not explicitly stated
Linton's theorem (equational theory)
Linton's_theorem_(equational_theory)
Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which
Graded_Lie_algebra
a symmetric sequence in a symmetric monoidal ∞-category C), an algebra over an operad, or O-algebra for short, is, roughly, a left module over O with
Operad_algebra
In mathematics, invertible homomorphism
unique. The term isomorphism is mainly used for algebraic structures and categories. In the case of algebraic structures, mappings are called homomorphisms
Isomorphism
Algebraic structure
algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are
Interior_algebra
French mathematician (1933–2015)
contributions to homological algebra, the theory of abelian categories, algebraic groups and the representation theory of algebras. Gabriel was born in the
Pierre_Gabriel
French mathematician (1928–2014)
of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory
Alexander_Grothendieck
free Lie algebra L ( X ) {\displaystyle L(X)} generated by X. In the language of category theory, the functor sending a set X to the Lie algebra generated
Free_Lie_algebra
Generalization of associativity properties
these operations. Given an operad O {\displaystyle O} , one defines an algebra over O {\displaystyle O} to be a set together with concrete operations
Operad
Left adjoint to a forgetful functor to sets
it relates to all types of algebraic structure (with finitary operations). It also has a formulation in terms of category theory, although this is in
Free_object
Associative algebra used in combinatorics
algebra and the incidence algebra are special cases of a category algebra, defined analogously; groups and posets being special kinds of categories.
Incidence_algebra
Category theory
In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli
Kleisli_category
Mathematical property
concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one
Semi-simplicity
quantum groups. Ringel (1990) generalized Hall algebras to more general categories, such as the category of representations of a quiver. A finite abelian
Hall_algebra
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
Algebraic structure in homological algebra
homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often
Differential_graded_algebra
Technical treatment of Boolean algebras
mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
Mathematical object
In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework
Initial_algebra
Various mathematical dualites
derived category of a symmetric algebra and that of an exterior algebra, as well as the BGG correspondence, which links the stable category of finite-dimensional
Koszul_duality
Algebraic topology uses abstract algebra to study topological spaces
theorem Abstract simplicial complex Simplicial set Simplicial category Chain (algebraic topology) Betti number Euler characteristic Genus Riemann–Hurwitz
List of algebraic topology topics
List_of_algebraic_topology_topics
Algebraic variety defined within an affine space
In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine
Affine_variety
Structure-preserving function between two rings
Bourbaki, N. (1998). Algebra I, Chapters 1–3. Springer. Eisenbud, David (1995). Commutative algebra with a view toward algebraic geometry. Graduate Texts
Ring_homomorphism
algebraic topology. By definition, the t-structure of a stable ∞-category is the t-structure of its homotopy category. Let C be a stable ∞-category with
Stable_∞-category
Type of category in mathematics
Order, Topology, Algebra, and Sheaf Theory. Cambridge University Press. ISBN 978-0-521-83414-8. Retrieved 4 April 2018. Extensive category at the nLab v
Extensive_category
Algebra used in 2D conformal field theories and string theory
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string
Vertex_operator_algebra
Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf
Braided_Hopf_algebra
Mathematical representation in functional analysis
representing commutative Banach algebras as algebras of continuous functions; the fact that for commutative C*-algebras, this representation is an isometric
Gelfand_representation
Branch of mathematics
operator-algebraic methods based on C*-algebras, von Neumann algebras, and spectral triples; algebraic approaches to noncommutative rings and graded algebras;
Noncommutative_geometry
In commutative algebra, an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field
Étale_algebra
Type of monoidal category in category theory
and category C {\displaystyle C} give rise to a category A l g P C {\displaystyle \mathrm {Alg} _{P}^{C}} of algebras whose objects are the algebras of
PROP_(category_theory)
free product of groups. In the category of commutative R-algebras, the free product of two algebras (in that category) is their tensor product. We first
Free product of associative algebras
Free_product_of_associative_algebras
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CATEGORY ALGEBRA
CATEGORY ALGEBRA
CATEGORY ALGEBRA
CATEGORY ALGEBRA
CATEGORY ALGEBRA
CATEGORY ALGEBRA
CATEGORY ALGEBRA
CATEGORY ALGEBRA
CATEGORY ALGEBRA
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