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:
“In so far as the statements of geometry speak about reality, they are not certain, and in so far as they are certain, they do not speak about reality.”
—Albert Einstein (1879–1955)
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“The statements of science are hearsay, reports from a world outside the world we know. What the poet tells us has long been known to us all, and forgotten. His knowledge is of our world, the world we are both doomed and privileged to live in, and it is a knowledge of ourselves, of the human condition, the human predicament.”
—John Hall Wheelock (1886–1978)