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:
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)