Definition
Let be an abelian category. We obtain the derived category in several steps:
- The basic object is the category of chain complexes in . Its objects will be the objects of the derived category but its morphisms will be altered.
- Pass to the homotopy category of chain complexes by identifying morphisms which are chain homotopic.
- Pass to the derived category by localizing at the set of quasi-isomorphisms. Morphisms in the derived category may be explicitly described as roofs, where s is a quasi-isomorphism and f is any morphism of chain complexes.
The second step may be bypassed since a homotopy equivalence is in particular a quasi-isomorphism. But then the simple roof definition of morphisms must be replaced by a more complicated one using finite strings of morphisms (technically, it is no longer a calculus of fractions), and the triangulated category structure of arises in the homotopy category. So the one step construction is more efficient in a way but more complicated and the result is less powerful.
Read more about this topic: Derived Category
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)