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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
In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i
Algebraic_topology_(object)
Branch of mathematics
proofs. Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. The basic goal is to find algebraic invariants
Topology
Topology on prime ideals and algebraic varieties
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different
Zariski_topology
Topological Object
Kleinian groups with the Chabauty topology. Algebraic topology (object) William Thurston, The geometry and topology of 3-manifolds, Princeton lecture
Geometric_topology_(object)
Mathematical structure
Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number
Grothendieck_topology
Mathematical object studied in the field of algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as
Algebraic_variety
Subject area in mathematics
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic
Algebraic_K-theory
Branch of mathematics
associative Outline of algebra Representation theory – Branch of mathematics that studies abstract algebraic structures Tensor – Algebraic object with geometric
Algebra
Generalization of algebraic spaces or schemes
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli
Algebraic_stack
Mathematical subject
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example
Combinatorial_topology
Branch of mathematics
traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end
Homological_algebra
Generalization of a scheme
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory
Algebraic_space
Mathematical space with a notion of closeness
compactness, and various separation axioms. For algebraic invariants see algebraic topology. Complete Heyting algebra – The system of all open sets of a given
Topological_space
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
Set with operations obeying given axioms
In mathematics, an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection
Algebraic_structure
Mathematics glossary
properties and concepts in algebraic topology in mathematics. See also: glossary of topology, list of algebraic topology topics, glossary of category
Glossary of algebraic topology
Glossary_of_algebraic_topology
In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also plays
Flat_topology
Branch of mathematics
Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in
Abstract_algebra
Type of optical illusion
objects mathematically using the algebraic topology concept of cohomology. An early example of an impossible object comes from Apolinère Enameled, a 1916 advertisement
Impossible_object
General theory of mathematical structures
work on algebraic topology. Category theory can be used in most areas of mathematics. In particular, many constructions of new mathematical objects from
Category_theory
Number of "holes" of a surface
projective algebraic scheme X {\displaystyle X} : the arithmetic genus and the geometric genus. When X {\displaystyle X} is an algebraic curve with field
Genus_(mathematics)
Topological space construction
(2008). Algebraic topology. Zürich: European mathematical society. ISBN 978-3-03719-048-7. Willard, Stephen (2004) [1970]. General Topology. Mineola
Quotient_space_(topology)
Type of Grothendieck topology on the category of schemes
algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology,
Étale_topology
linear partial differential equations, it is a branch of algebraic geometry and algebraic topology that uses methods from sheaf theory and complex analysis
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Mathematical approach
topology can be viewed from an algebraic point of view (lattice-theoretic). Karl Menger was an early pioneer in the field, and his work on topology without
Pointless_topology
lattice Algebraic poset Scott domain Algebraic lattice Scott information system Powerdomain Scott topology Scott continuity Lindenbaum algebra Zorn's lemma
List_of_order_theory_topics
Certain generalizations of groups
the axioms for group objects. Here 0 is the initial object of C. Cogroup objects occur naturally in algebraic topology. Hopf algebras can be seen as a generalization
Group_object
duality in algebraic topology" (PDF). In James, I.M. (ed.). History of topology. North Holland. pp. 725–745. ISBN 978-0-444-82375-5. dual object in a closed
Dual_object
Branch of mathematics
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts
Derived_algebraic_geometry
Branch of mathematics
captured algebraically, differential topology has strong links to algebraic topology. The central goal of the field of differential topology is the classification
Differential_topology
Categorical generalization of a function space in set theory
1007/978-1-4757-4721-8_5. ISBN 978-0387984032. Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for proof
Exponential_object
Generalisation of a sheaf; a fibered category that admits effective descent
these, while algebraic stacks are locally affine in the smooth topology so one can use the smooth topology in this case. For general algebraic stacks the
Stack_(mathematics)
Two-dimensional manifold
context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry
Surface_(topology)
Subsets whose union equals the whole set
Geometrical object Willard, Stephen (1998). General Topology. Dover Publications. p. 104. ISBN 0-486-43479-6. Bott, Tu (1982). Differential Forms in Algebraic Topology
Cover_(topology)
areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric
Glossary_of_general_topology
Branch of algebra that studies commutative rings
ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings
Commutative_algebra
Mathematical object in category theory
and also the various cohomology theories in group theory, algebraic topology and algebraic geometry. The categories being used are typically functor categories
Injective_object
Branch of functional analysis
algebras, many limit algebras. Banach algebra – Particular kind of algebraic structure Matrix mechanics – Formulation of quantum mechanics Topologies
Operator_algebra
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Mathematical concept
inspired by (and generalized from) the concept of compactness in topology. An object X in a category C which admits all filtered colimits (also known
Compact_object_(mathematics)
Area of combinatorics
problems in algebra. The term "algebraic combinatorics" was introduced in the late 1970s. Through the early or mid-1990s, typical combinatorial objects of interest
Algebraic_combinatorics
Branch of topology
branches of topology, including differential topology, geometric topology, and algebraic topology. The fundamental concepts in point-set topology are continuity
General_topology
Mathematical concept
the dual of any finite-dimensional full matrix algebra Mn(C) consists of a single point. The topology of  can be defined in several equivalent ways.
Spectrum_of_a_C*-algebra
Representation of mathematical space
applications both in and outside of mathematics, for instance in algebraic topology, in complex analysis, and in modeling. On the one hand, it is sometimes
Triangulation_(topology)
Structure in algebraic geometry
In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes
Nisnevich_topology
Branch of mathematics
the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial
Geometry
Algebraic concept in measure theory, also referred to as an algebra of sets
Alexandrov topology – Type of topology in mathematics Algebra of sets – Identities and relationships involving sets Boolean ring – Algebraic structure
Field_of_sets
Collection of open sets used to define a topology
a basic open set. The Zariski topology of C n {\displaystyle \mathbb {C} ^{n}} is the topology that has the algebraic sets as closed sets. It has a base
Base_(topology)
emergence of abstract algebra. This approach explored the axiomatic basis of arbitrary algebraic operations. The invention of new algebraic systems based on
History_of_algebra
Area of mathematics using condensed sets
aims to unify various mathematical subfields, including topology, complex geometry, and algebraic geometry.[citation needed] In particular, Kiran Kedlaya
Condensed_mathematics
Branch of mathematics
commutative algebra, and optimization. Nonlinear algebra is closely related to algebraic geometry, where the main objects of study include algebraic equations
Nonlinear_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
Topological spaces whose union is a boundary
cobordism. Cobordisms are central objects of study in geometric topology and algebraic topology. In geometric topology, cobordisms are intimately connected
Cobordism
Set of a ring's prime ideals
then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed
Spectrum_of_a_ring
Algebraic structure used in logic
complete Heyting algebra. Complete Heyting algebras thus become a central object of study in pointless topology. Every Heyting algebra whose set of non-greatest
Heyting_algebra
Branch of mathematics studying (smooth) functions of manifolds
manifold into another. Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens
Geometric_topology
Algebraic structure with addition and multiplication
influenced by problems and ideas of algebraic number theory and algebraic geometry. In turn, commutative algebra is a fundamental tool in these branches
Ring_(mathematics)
mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are
List_of_topology_topics
Generalization of algebraic variety
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of an algebraic variety in several ways, such as taking
Scheme_(mathematics)
Property of a mathematical space
University Press. p. 24. ISBN 978-1-4008-7566-5. Extract of page 24 "Algebraic Topology" (PDF). Cornell University. Retrieved 2026-05-19. Fractal Dimension
Dimension
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
Unsolved problem in geometry
unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties
Hodge_conjecture
Construction in algebra
trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space
Hopf_algebra
Two closely related mathematical subjects
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic
Algebraic geometry and analytic geometry
Algebraic_geometry_and_analytic_geometry
Additional mathematical object
induce its topology. Its order and algebraic structure make it into an ordered field. Its algebraic structure and topology make it into a Lie group, a type
Mathematical_structure
In algebraic geometry, the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. It combines several
H_topology
Branch of mathematics
generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of closed model categories. This subject
Homotopical_algebra
In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts
Directed_algebraic_topology
Concept in mathematics
intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology. A one-dimensional formal
Formal_group_law
Algebraic variety with a group structure
mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus
Algebraic_group
In algebraic topology, an S {\displaystyle \mathbb {S} } -object (also called a symmetric sequence) is a sequence { X ( n ) } {\displaystyle \{X(n)\}}
S-object
Topological invariant in mathematics
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré
Euler_characteristic
Mathematic concept
beginning of this article. In algebraic geometry, a symmetric power is defined in a way similar to that in algebraic topology. For example, if X = Spec
Symmetric_power
Generalization of category theory
category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as the fundamental
Higher_category_theory
*-algebra of bounded operators on a Hilbert space
a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the
Von_Neumann_algebra
All points and limit points in a subset of a topological space
In topology, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of
Closure_(topology)
French mathematician (1928–2014)
which use algebraic techniques to study topological objects, has influenced the development of algebraic number theory, algebraic topology, and representation
Alexander_Grothendieck
Long exact sequence
In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space
Gysin_homomorphism
persistence modules have been one of the primary algebraic structures studied in the field of applied topology. Let T {\displaystyle T} be a totally ordered
Persistence_module
Branch of mathematics
structures that are often specified via algebraic operations and defining identities are Heyting algebras and Boolean algebras, which both introduce a new operation
Order_theory
talk, where he described it as: The ultimate object of algebraic homotopy is to construct a purely algebraic theory, which is equivalent to homotopy theory
Algebraic_homotopy
Computable topology is a discipline in mathematics that studies the topological and algebraic structure of computation. Computable topology is not to be
Computable_topology
Tool to track locally defined data attached to the open sets of a topological space
nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such
Sheaf_(mathematics)
Topological construction on a map between spaces
Cofibration Mapping cone (homological algebra) Rotman, Joseph J. (1988). An Introduction to Algebraic Topology. Springer-Verlag. ISBN 0-387-96678-1. See
Mapping_cone_(topology)
Describes the fundamental group in terms of a cover by two open path-connected subspaces
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's
Seifert–Van_Kampen_theorem
In algebraic topology, the nth symmetric product of a topological space consists of the unordered n-tuples of its elements. If one fixes a basepoint, there
Symmetric_product_(topology)
Mathematical construction used in homotopy theory
introduction to simplicial sets" (PDF). May, J. Peter. Simplicial Objects in Algebraic Topology, University of Chicago Press 1967 simplicial set at the nLab
Simplicial_set
an initial object ∗ {\displaystyle *} . These are useful constructions because they help export some of the ideas from algebraic topology and homotopy
H-object
Special objects used in (mathematical) category theory
theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism
Initial_and_terminal_objects
Topics referred to by the same term
Quotient space (topology), in case of topological spaces Quotient space (linear algebra), in case of vector spaces Quotient space of an algebraic stack Quotient
Quotient_space
How spheres of various dimensions can wrap around each other
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other.
Homotopy_groups_of_spheres
Banach space of a dual
predual A ∗ {\displaystyle A_{*}} induces a weak topology on A {\displaystyle A} , under which algebra operations are separately weak continuous. Ruan
Predual
Construct in mathematics
especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry
Gerbe
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
Form taken by the network of interconnections of a circuit
high-pass and low-pass topologies even though the network topology is identical. A more correct term for these classes of object (that is, a network where
Circuit_topology_(electrical)
generalization of the corresponding notion (stabilization (topology)) in classical algebraic topology. By definition, the t-structure of a stable ∞-category
Stable_∞-category
1957 mathematics paper by Alexander Grothendieck
It revolutionized the subject of homological algebra, a purely algebraic aspect of algebraic topology. It removed the need to distinguish the cases of
Grothendieck's_Tôhoku_paper
Branch of mathematics that studies the properties of groups
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known
Group_theory
Algebraic structure used in topology
theory and algebraic topology, cohomology is a way of attaching algebraic invariants to a topological space or other mathematical object that encode
Cohomology
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ALGEBRAIC TOPOLOGY-OBJECT
ALGEBRAIC TOPOLOGY-OBJECT
Surname or Lastname
English (also common in Wales)
English (also common in Wales) : patronymic from Edward.One of the earliest American bearers of this very common English surname was William Edwards, the son of Rev. Richard Edwards, a London clergyman in the age of Elizabeth I, who came to New England about 1640. His descendant Jonathan (1703–58), of East Windsor, CT, was a prominent Congregational clergyman whose New England theology led to the first Great Awakening, a great religious revival.
Boy/Male
Muslim
Intended, Aimed at, Object, Proposed
Boy/Male
Tamil
Decorated, An object that gives light, And never stops doing so
Boy/Male
Muslim
Objective, Goal
Boy/Male
Tamil
Object in the Sky cloud, Moon
Surname or Lastname
English
English : nickname for a foolish or eccentric person, from a diminutive of Foll, from Old French fol ‘mad’, ‘stupid’ (Late Latin follis, originally a noun denoting any of various objects filled with air, but later transferred to vain and empty-headed notions).
Surname or Lastname
English (of Norman origin) and French
English (of Norman origin) and French : occupational name for a maker of glass objects, Old French verrie(o)r (from verre, voir(r)e ‘glass’, Latin vitrum).
Girl/Female
Muslim
Rarity, Rare object, Novelty
Boy/Male
Hindu
Object in the Sky cloud, Moon
Surname or Lastname
English
English : variant of Styles.German : topographic name for someone who lived on or by a hill, from Middle High German stickel ‘hill’, ‘slope’.German : nickname from Middle High German stickel ‘prickle’, ‘spine’, ‘pointed object’.
Surname or Lastname
English
English : occupational name for a maker of dowels and similar objects, from an agent derivative of Middle English dowle ‘dowel’, ‘headless peg’, ‘bolt’.
Surname or Lastname
French
French : metonymic occupational name for a gardener, from the objective case (gard) of Old French gardin ‘garden’.English : variant spelling of Guard.Norwegian : habitational name from a farmstead so named, from Old Norse garðr ‘farm’.Swedish (Gård) : topographic or ornamental name from gård ‘farm’.
Surname or Lastname
English (mainly Newcastle and Durham)
English (mainly Newcastle and Durham) : of uncertain origin, probably a derivative of northern Middle English stang ‘pole’ (of Old Norse origin). Possible meanings include a topographic name for someone who lived by a pole or stake (compare Stakes) or an occupational name for someone armed with one. Alternatively, it may be a nickname for someone who had ‘ridden the stang’, i.e. been carried on a pole through the streets as an object of derision, in punishment for some misdemeanor. However, this custom is of uncertain antiquity.Orcadian : probably a habitational name from a minor place called Stanagar in the parish of Stromness.German : occupational name for a maker of shafts for spears and the like, from an agent derivative of Middle High German stange ‘pole’, ‘shaft’.
Boy/Male
Tamil
Object in the Sky cloud, Moon
Boy/Male
Muslim
Intended, Aimed at, Object, Proposed
Boy/Male
Tamil
Decorated, An object that gives light, And never stops doing so
Surname or Lastname
English and Scottish
English and Scottish : occupational name for a maker of objects of wood, metal, or bone by turning on a lathe, from Anglo-Norman French torner (Old French tornier, Latin tornarius, a derivative of tornus ‘lathe’). The surname may also derive from any of various other senses of Middle English turn, for example a turnspit, a translator or interpreter, or a tumbler.English : nickname for a fast runner, from Middle English turnen ‘to turn’ + ‘hare’.English : occupational name for an official in charge of a tournament, Old French tornei (in origin akin to 1).Jewish (eastern Ashkenazic) : habitational name from a place called Turno or Turna, in Poland and Belarus, or from the city of Tarnów (Yiddish Turne) in Poland.Translated or Americanized form of any of various other like-meaning or like-sounding Jewish surnames.South German (T(h)ürner) : occupational name for a guard in a tower or a topographic name from Middle High German turn ‘tower’, or a habitational name for someone from any of various places named Thurn, for example in Austria.
Girl/Female
African, Arabic, Swahili
Apology; Virgin
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : from the Old French personal name Reinger, Rainger, composed of the Germanic elements ragin ‘advice’, ‘counsel’ + gÄr, gÄ“r ‘spear’, ‘lance’.English : occupational name for a maker of rings (see Ring 1) or for a bell ringer, from Middle English ring(en) ‘to ring’, Old English hringan.German : occupational name for a turner, someone who made objects by rotating them on a lathe or wheel.
Boy/Male
Tamil
Decorated, An object that gives light, And never stops doing so
ALGEBRAIC TOPOLOGY-OBJECT
ALGEBRAIC TOPOLOGY-OBJECT
ALGEBRAIC TOPOLOGY-OBJECT
ALGEBRAIC TOPOLOGY-OBJECT
ALGEBRAIC TOPOLOGY-OBJECT
ALGEBRAIC TOPOLOGY-OBJECT
ALGEBRAIC TOPOLOGY-OBJECT
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