# Category: Lemmas in category theory

Zig-zag lemma
In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in
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 homologi
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.It states that for the following commutative
Yoneda lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation o
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.The five lemma is not
Horseshoe lemma
In homological algebra, the horseshoe lemma, also called the simultaneous resolution theorem, is a statement relating resolutions of two objects and to resolutions ofextensions of by . It says that if
Splitting lemma
In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence 1. * Left spl
Nine lemma
In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. It states: if the diagram to the r