• topology glossary
  Topology glossary
  This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology...
  
  
• List of topology topics
• List of general topology topics
• List of geometric topology topics
• Publications in topology

• Topological property
  Topological property
  In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic...
  
  

Homology (mathematics)

Homology (mathematics)

In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...

• Simplex
  Simplex
  In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
  
  
• Simplicial complex
  Simplicial complex
  In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
  
  
  • Polytope
    Polytope
    In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
    
    
  • Triangulation
    Triangulation
    In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...
    
    
  • Barycentric subdivision
    Barycentric subdivision
    In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.The name...
    
    
  • Simplicial approximation theorem
    Simplicial approximation theorem
    In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices — that is,...
    
    
  • Abstract simplicial complex
    Abstract simplicial complex
    In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of finite sets closed under the operation of taking subsets...
    
    
  • Simplicial set
    Simplicial set
    In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space...
    
    
  • Simplicial category
    Simplicial category
    In mathematics, the simplex category is the category of finite ordinals and order preserving maps. It is used to define simplicial and cosimplicial objects.-Formal definition:...
    
    
• Chain (algebraic topology)
  Chain (algebraic topology)
  In algebraic topology, a simplicial k-chainis a formal linear combination of k-simplices.-Integration on chains:Integration is defined on chains by taking the linear combination of integrals over the simplices in the chain with coefficients typically integers.The set of all k-chains forms a group...
  
  
• Betti number
  Betti number
  In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
  
  
• Euler characteristic
  Euler characteristic
  In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
  
  
  • Genus
    Genus (mathematics)
    In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
    
    
  • Riemann-Hurwitz formula
    Riemann-Hurwitz formula
    In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case...
    
    
• Singular homology
  Singular homology
  In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
  
  
• Cellular homology
  Cellular homology
  In mathematics, cellular homology in algebraic topology is a homology theory for CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.- Definition :...
  
  
• Relative homology
  Relative homology
  In algebraic topology, a branch of mathematics, the homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways...
  
  
• Mayer–Vietoris sequence
• Excision theorem
  Excision theorem
  In algebraic topology, a branch of mathematics, the excision theorem is a useful theorem about relative homology—given topological spaces X and subspaces A and U such that U is also a subspace of A, the theorem says that under certain circumstances, we can cut out U from both spaces such...
  
  
• Universal coefficient theorem
  Universal coefficient theorem
  In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A...
  
  
• Cohomology
  Cohomology
  In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
  
  
  • List of cohomology theories
  • Cocycle class
    Cocycle class
    In mathematics, more specifically in cohomology theory, a q \, \!-cocycle \alpha \, \! in the cochain group S^q \, \! is associated with a unique equivalence class [\alpha] \in H^q \, \! known as the cocycle class or coclass of \alpha. \, \!...
    
    
  • Cup product
    Cup product
    In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...
    
    
  • Cohomology ring
    Cohomology ring
    In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is...
    
    
  • De Rham cohomology
    De Rham cohomology
    In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
    
    
  • Čech cohomology
    Cech cohomology
    In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.-Motivation:...
    
    
  • Alexander-Spanier cohomology
    Alexander-Spanier cohomology
    In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomologytheory for topological spaces, introduced by for the special case of compact metric spaces, and by for all topological spaces, based on a suggestion of A. D...
    
    
  • Intersection cohomology
    Intersection cohomology
    In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years.Intersection cohomology was...
    
    
  • Lyusternik-Schnirelmann category
• Poincaré duality
  Poincaré duality
  In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
  
  
• Fundamental class
  Fundamental class
  In mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold"...
  
  
• Applications
  • Jordan curve theorem
    Jordan curve theorem
    In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...
    
    
  • Brouwer fixed point theorem
    Brouwer fixed point theorem
    Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
    
    
  • Invariance of domain
    Invariance of domain
    Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states:The theorem and its proof are due to L.E.J. Brouwer, published in 1912...
    
    
  • Lefschetz fixed-point theorem
    Lefschetz fixed-point theorem
    In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X...
    
    
  • Hairy ball theorem
    Hairy ball theorem
    The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on an even-dimensional n-sphere. An ordinary sphere is a 2-sphere, so that this theorem will hold for an ordinary sphere...
    
    
  • Degree (mathematics)
    Degree (mathematics)
    In mathematics, there are several meanings of degree depending on the subject.- Unit of angle :A degree , usually denoted by ° , is a measurement of a plane angle, representing 1⁄360 of a turn...
    
    
  • Borsuk-Ulam Theorem
  • Ham sandwich theorem
    Ham sandwich theorem
    In measure theory, a branch of mathematics, the ham sandwich theorem, also called the Stone–Tukey theorem after Arthur H. Stone and John Tukey, states that given measurable "objects" in -dimensional space, it is possible to divide all of them in half with a single -dimensional hyperplane...
    
    
  • Homology sphere
    Homology sphere
    In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is,andTherefore X is a connected space, with one non-zero higher Betti number: bn...
    
    

Homotopy theory

• Homotopy
  Homotopy
  In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
  
  
• Path (topology)
  Path (topology)
  In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
  
  
• Fundamental group
  Fundamental group
  In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
  
  
• Homotopy group
  Homotopy group
  In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
  
  
• Seifert–van Kampen theorem
  Seifert–van Kampen theorem
  In mathematics, the Seifert-van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path-connected subspaces U and V that cover X...
  
  
• Pointed space
  Pointed space
  In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
  
  
• Winding number
  Winding number
  In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
  
  
• Simply connected
  • Universal cover
• Monodromy
  Monodromy
  In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
  
  
• Homotopy lifting property
  Homotopy lifting property
  In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space E to another one, B...
  
  
• Mapping cylinder
  Mapping cylinder
  In mathematics, specifically algebraic topology, the mapping cylinder of a function f between topological spaces X and Y is the quotientM_f = \,/\,\sim...
  
  
• Mapping cone
  Mapping cone
  In mathematics, especially homotopy theory, the mapping cone is a construction C_f of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated Cf.-Definition:...
  
  
• Wedge sum
  Wedge sum
  In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces the wedge sum of X and Y is the quotient of the disjoint union of X and Y by the identification x0 ∼ y0:X\vee Y = \;/ \sim,\,where ∼ is the...
  
  
• Smash product
  Smash product
  In mathematics, the smash product of two pointed spaces X and Y is the quotient of the product space X × Y under the identifications  ∼  for all x ∈ X and y ∈ Y. The smash product is usually denoted X ∧ Y...
  
  
• Adjunction space
  Adjunction space
  In mathematics, an adjunction space is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be a topological spaces with A a subspace of Y. Let f : A → X be a continuous map...
  
  
• Cohomotopy
• Cohomotopy group
  Cohomotopy group
  In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and point-preserving continuous maps to the category of sets and functions...
  
  
• Brown's representability theorem
  Brown's representability theorem
  In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions on a contravariant functor F on the homotopy category Hot of pointed CW complexes, to the category of sets Set, to be a representable functor...
  
  
• Eilenberg–MacLane space
• Fibre bundle
  • Möbius strip
    Möbius strip
    The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
    
    
  • Line bundle
    Line bundle
    In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
    
    
    • Canonical line bundle
      Canonical line bundle
      The canonical or tautological line bundle on a projective space appears frequently in mathematics, often in the study of characteristic classes...
      
      
  • Vector bundle
    Vector bundle
    In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
    
    
  • Associated bundle
    Associated bundle
    In mathematics, the theory of fiber bundles with a structure group G allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with a group action of G...
    
    
  • Fibration
    Fibration
    In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
    
    
  • Hopf bundle
    Hopf bundle
    In the mathematical field of topology, the Hopf fibration describes a 3-sphere in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle...
    
    
  • Classifying space
    Classifying space
    In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
    
    
• Cofibration
  Cofibration
  In mathematics, in particular homotopy theory, a continuous mappingi\colon A \to X,where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines...
  
  
• Homotopy groups of spheres
  Homotopy groups of spheres
  In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...
  
  
• Plus construction
  Plus construction
  In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups. It was introduced by Daniel Quillen. Given a perfect normal subgroup of the fundamental group of a connected CW complex X, attach two-cells along...
  
  
• Whitehead theorem
  Whitehead theorem
  In homotopy theory , the Whitehead theorem states that if a continuous mapping f between topological spaces X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence provided X and Y are connected and have the homotopy-type of CW complexes. This result was proved by J....
  
  
• Weak equivalence
  Weak equivalence
  -Mathematics:In mathematics, a weak equivalence is a notion from homotopy theory which in some sense identifies objects that have the same basic "shape"...
  
  
• Hurewicz theorem
  Hurewicz theorem
  In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism...
  
  
• H-space
  H-space
  In mathematics, an H-space is a topological space X together with a continuous map μ : X × X → X with an identity element e so that μ = μ = x for all x in X...
  
  

Further developments

• Künneth theorem
  Künneth theorem
  In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y and their...
  
  
• De Rham cohomology
  De Rham cohomology
  In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
  
  
• Obstruction theory
  Obstruction theory
  In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.-In homotopy theory:...
  
  
• Characteristic class
  Characteristic class
  In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
  
  
  • Chern class
    Chern class
    In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...
    
    
  • Chern-Simons form
    Chern-Simons form
    In mathematics, the Chern–Simons forms are certain secondary characteristic classes. They have been found to be of interest in gauge theory, and they define the action of Chern–Simons theory...
    
    
  • Pontryagin class
    Pontryagin class
    In mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four...
    
    
  • Pontryagin number
  • Stiefel-Whitney class
• Poincaré conjecture
  Poincaré conjecture
  In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
  
  
• Cohomology operation
  Cohomology operation
  In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from...
  
  
  • Steenrod algebra
    Steenrod algebra
    In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.For a given prime number p, the Steenrod algebra Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p...
    
    
• Bott periodicity theorem
  Bott periodicity theorem
  In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy...
  
  
• K-theory
  K-theory
  In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
  
  
  • Topological K-theory
    Topological K-theory
    In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as K-theory that were introduced by Alexander Grothendieck...
    
    
  • Adams operation
    Adams operation
    In mathematics, an Adams operationis a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams...
    
    
  • Algebraic K-theory
    Algebraic K-theory
    In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
    
    
  • Whitehead torsion
    Whitehead torsion
    In geometric topology, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau, which is an element in the Whitehead group Wh. These are named after the mathematician J. H. C...
    
    
  • Twisted K-theory
    Twisted K-theory
    In mathematics, twisted K-theory is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory....
    
    
• Cobordism
  Cobordism
  In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
  
  
• Thom space
  Thom space
  In mathematics, the Thom space, Thom complex, or Pontryagin-Thom construction of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space....
  
  
• Suspension functor
• Stable homotopy theory
  Stable homotopy theory
  In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
  
  
• Spectrum (homotopy theory)
  Spectrum (homotopy theory)
  In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, any of which gives a context for the same stable homotopy theory....
  
  
• Morava K-theory
  Morava K-theory
  In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s...
  
  
• Hodge conjecture
  Hodge conjecture
  The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
  
  
• Weil conjectures
  Weil conjectures
  In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
  
  
• Directed algebraic topology
  Directed algebraic topology
  In mathematics, directed algebraic topology is a form of algebraic topology that studies topological spaces equipped with a family of directed paths, closed under some operations....
  
  

Homological algebra

Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

• Chain complex
  Chain complex
  In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
  
  
• Commutative diagram
  Commutative diagram
  In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
  
  
• Exact sequence
  Exact sequence
  An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
  
  
  • Five lemma
    Five lemma
    In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams....
    
    
  • Short five lemma
    Short five lemma
    In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma....
    
    
  • Snake lemma
    Snake lemma
    The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology...
    
    
  • Splitting lemma
    Splitting lemma
    In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent....
    
    
  • Extension problem
• Spectral sequence
  Spectral sequence
  In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
  
  
• Abelian category
  Abelian category
  In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
  
  
• Group cohomology
  Group cohomology
  In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
  
  
• Sheaf
  Sheaf (mathematics)
  In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
  
  
  • Sheaf cohomology
    Sheaf cohomology
    In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
    
    
  • Grothendieck topology
    Grothendieck topology
    In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
    
    
• Derived category
  Derived category
  In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...