Derived category
Remarks
Remarks
-------

For certain purposes (see below) one uses *bounded\-below* (X^n \= 0 for n \\ll 0), *bounded\-above* (X^n \= 0 for n \\gg 0) or *bounded* (X^n \= 0 for \|n\| \\gg 0) complexes instead of unbounded ones. The corresponding derived categories are usually denoted *D\+(A)*, *D−(A)* and *Db(A)*, respectively.

If one adopts the classical point of view on categories, that there is a [set](/wiki/Set_%28mathematics%29 "Set (mathematics)") of morphisms from one object to another (not just a [class](/wiki/Class_%28set_theory%29 "Class (set theory)")), then one has to give an additional argument to prove this. If, for example, the abelian category *A* is small, i.e. has only a set of objects, then this issue will be no problem. Also, if *A* is a [Grothendieck abelian category](/wiki/Grothendieck_category "Grothendieck category"), then the derived category *D*(*A*) is equivalent to a full subcategory of the homotopy category *K*(*A*), and hence has only a set of morphisms from one object to another. Grothendieck abelian categories include the category of modules over a ring, the category of sheaves of abelian groups on a topological space, and many other examples.

Composition of morphisms, i.e. roofs, in the derived category is accomplished by finding a third roof on top of the two roofs to be composed. It may be checked that this is possible and gives a well\-defined, associative composition.

Since *K(A)* is a [triangulated category](/wiki/Triangulated_category "Triangulated category"), its localization *D(A)* is also triangulated. For an integer *n* and a complex *X*, define the complex *X*\[*n*] to be *X* shifted down by *n*, so that 
X\[n]^{i} \= X^{n\+i},
with differential
d\_{X\[n]} \= (\-1\)^n d\_X.
By definition, a distinguished triangle in *D(A)* is a triangle that is isomorphic in *D(A)* to the triangle *X* → *Y* → Cone(*f*) → *X*\[1] for some map of complexes *f*: *X* → *Y*. Here Cone(*f*) denotes the [mapping cone](/wiki/Mapping_cone_%28homological_algebra%29 "Mapping cone (homological algebra)") of *f*. In particular, for a short exact sequence 
0 \\rightarrow X \\rightarrow Y \\rightarrow Z \\rightarrow 0
in *A*, the triangle *X* → *Y* → *Z* → *X*\[1] is distinguished in *D(A)*. Verdier explained that the definition of the shift *X*\[1] is forced by requiring *X*\[1] to be the cone of the morphism *X* → 0\.

By viewing an object of *A* as a complex concentrated in degree zero, the derived category *D(A)* contains *A* as a [full subcategory](/wiki/Subcategory "Subcategory"). Morphisms in the derived category include information about all [Ext groups](/wiki/Ext_functor "Ext functor"): for any objects *X* and *Y* in *A* and any integer *j*,

\\text{Hom}\_{D(\\mathcal{A})}(X,Y\[j]) \= \\text{Ext}^j\_{\\mathcal{A(X,Y).

When does the issue of having a set of morphisms from one object to another become no problem?