Remarks
Two chain homotopic maps f and g induce the same maps on homology because (f − g) sends cycles to boundaries, which are zero in homology. In particular a homotopy equivalence is a quasi-isomorphism. (The converse is false in general.) This shows that there is a canonical functor to the derived category (if A is abelian).
Read more about this topic: Homotopy Category Of Chain Complexes
Famous quotes containing the word remarks:
“There are remarks that sow and remarks that reap.”
—Ludwig Wittgenstein (1889–1951)
“So, too, if, to our surprise, we should meet one of these morons whose remarks are so conspicuous a part of the folklore of the world of the radio—remarks made without using either the tongue or the brain, spouted much like the spoutings of small whales—we should recognize him as below the level of nature but not as below the level of the imagination.”
—Wallace Stevens (1879–1955)
“I thought my razor was dull until I heard his speech and that reminds me of a story that’s so dirty I’m ashamed to think of it myself.”
—S.J. Perelman, U.S. screenwriter, Bert Kalmar, Harry Ruby, and Norman Z. McLeod. Groucho Marx, Horsefeathers, as a newly-appointed college president commenting on the remarks of Huxley College’s outgoing president (1932)