mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, especially

homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

and other applications of

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 prototypical example of an abelian category is the category o ...

theory, the short five lemma is a special case of the

five lemma In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma (mathematics), lemma about commutative diagrams. The five lemma is not only valid for abelian cat ...

. It states that for the following

commutative diagram 350px, 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 s ...

(in any abelian

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

, or in the

category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories The ...

), if the rows are short exact sequences, and if ''g'' and ''h'' are

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

s, then ''f'' is an isomorphism as well. : It follows immediately from the

. The essence of the lemma can be summarized as follows: if you have a

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

''f'' from an object ''B'' to an object ', and this homomorphism induces an isomorphism from a subobject ''A'' of ''B'' to a subobject ' of ' and also an isomorphism from the factor object ''B''/''A'' to '/', then ''f'' itself is an isomorphism. Note however that the existence of ''f'' (such that the diagram commutes) has to be assumed from the start; two objects ''B'' and ' that simply have isomorphic sub- and factor objects need not themselves be isomorphic (for example, in the

category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object o ...

, ''B'' could be the

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

of order four and ' the

Klein four-group In mathematics, the Klein four-group is an abelian group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identi ...

References

* * {{DEFAULTSORT:Short Five Lemma Homological algebra Lemmas in category theory