A derived category of an Abelian category 
is the category 
obtained from the homotopy
category of complexes by formally inverting quasi-isomorphisms.
A quasi-isomorphism is a map of complexes inducing
isomorphisms on all cohomology
objects. Thus distinct complexes can represent the same derived object even when
they are not isomorphic term by term (Stacks Project n.d.).
The bounded derived category 
restricts to bounded complexes. It is triangulated, with
shift of complexes and distinguished triangles represented by mapping cones. For
a projective variety 
, the notation 
uses the Abelian category
of coherent sheaves. For example, an exact
sequence of sheaves gives a distinguished triangle
after viewing each sheaf as a complex concentrated in degree
zero.
