Faster access than browser!
29 relations: Abelian category, Abelian group, Category of groups, Commutative diagram, Dual (category theory), Epimorphism, Exact sequence, Field (mathematics), Function (mathematics), Group (mathematics), Homological algebra, Homology (mathematics), Homomorphism, If and only if, Image (mathematics), Injective function, Isomorphism, Kernel (algebra), Lemma (mathematics), Mathematics, Mitchell's embedding theorem, Module (mathematics), Monomorphism, Nine lemma, Ring (mathematics), Short five lemma, Snake lemma, Surjective function, Vector space.
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.
New!!: Five lemma and Abelian category · See more »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
New!!: Five lemma and Abelian group · See more »
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms.
New!!: Five lemma and Category of groups · See more »
Commutative diagram
The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result.
New!!: Five lemma and Commutative diagram · See more »
Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop.
New!!: Five lemma and Dual (category theory) · See more »
Epimorphism
In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f: X → Y that is right-cancellative in the sense that, for all morphisms, Epimorphisms are categorical analogues of surjective functions (and in the category of sets the concept corresponds to the surjective functions), but it may not exactly coincide in all contexts; for example, the inclusion \mathbb\to\mathbb is a ring-epimorphism.
New!!: Five lemma and Epimorphism · See more »
Exact sequence
An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry.
New!!: Five lemma and Exact sequence · See more »
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
New!!: Five lemma and Field (mathematics) · See more »
Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
New!!: Five lemma and Function (mathematics) · See more »
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
New!!: Five lemma and Group (mathematics) · See more »
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.
New!!: Five lemma and Homological algebra · See more »
Homology (mathematics)
In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
New!!: Five lemma and Homology (mathematics) · See more »
Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
New!!: Five lemma and Homomorphism · See more »
If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
New!!: Five lemma and If and only if · See more »
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.
New!!: Five lemma and Image (mathematics) · See more »
Injective function
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.
New!!: Five lemma and Injective function · See more »
Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
New!!: Five lemma and Isomorphism · See more »
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.
New!!: Five lemma and Kernel (algebra) · See more »
Lemma (mathematics)
In mathematics, a "helping theorem" or lemma (plural lemmas or lemmata) is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself.
New!!: Five lemma and Lemma (mathematics) · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Five lemma and Mathematics · See more »
Mitchell's embedding theorem
Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules.
New!!: Five lemma and Mitchell's embedding theorem · See more »
Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
New!!: Five lemma and Module (mathematics) · See more »
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism.
New!!: Five lemma and Monomorphism · See more »
Nine lemma
In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups.
New!!: Five lemma and Nine lemma · See more »
Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
New!!: Five lemma and Ring (mathematics) · See more »
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.
New!!: Five lemma and Short five lemma · See more »
Snake lemma
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences.
New!!: Five lemma and Snake lemma · See more »
Surjective function
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).
New!!: Five lemma and Surjective function · See more »
Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
New!!: Five lemma and Vector space · See more »
Redirects here:
5 lemma, 5-lemma, Five Lemma, Five-lemma, Four lemma, Steenrod five lemma, The five lemma.
References
[1] https://en.wikipedia.org/wiki/Five_lemma
