Statements
Consider the following commutative diagram in any abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or in the category of groups.
The five lemma states that, if the rows are exact, m and p are isomorphisms, l is an epimorphism, and q is a monomorphism, then n is also an isomorphism.
The two four-lemmas state:
(1) If the rows in the commutative diagram
are exact and m and p are epimorphisms and q is a monomorphism, then n is an epimorphism.
(2) If the rows in the commutative diagram
are exact and m and p are monomorphisms and l is an epimorphism, then n is a monomorphism.
Read more about this topic: Five Lemma
Famous quotes containing the word statements:
“We assume that politicians are without honor. We read their statements trying to crack the code. The scandals of their politics: not so much that men in high places lie, only that they do so with such indifference, so endlessly, still expecting to be believed. We are accustomed to the contempt inherent in the political lie.”
—Adrienne Rich (b. 1929)
“Dogmatic theological statements are neither logical propositions nor poetic utterances. They are “shaggy dog” stories; they have a point, but he who tries too hard to get it will miss it.”
—W.H. (Wystan Hugh)
“Is it true or false that Belfast is north of London? That the galaxy is the shape of a fried egg? That Beethoven was a drunkard? That Wellington won the battle of Waterloo? There are various degrees and dimensions of success in making statements: the statements fit the facts always more or less loosely, in different ways on different occasions for different intents and purposes.”
—J.L. (John Langshaw)