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:
“An illustrious individual remarks that Mrs. [Elizabeth Cady] Stanton is the salt, Anna Dickinson the pepper, and Miss [Susan B.] Anthony the vinegar of the Female Suffrage movement. The very elements get the “white male” into a nice pickle.”
—Anonymous, U.S. women’s magazine contributor. The Revolution (August 19, 1869)
“There are remarks that sow and remarks that reap.”
—Ludwig Wittgenstein (1889–1951)
“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)