This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
The concept of derived functors explains and clarifies many of these observations.
Suppose we are given a covariant left exact functor F : A → B between two abelian categories A and B.
Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right.
The crucial assumption we need to make about our abelian category A is that it has enough injectives, meaning that for every object A in A there exists a monomorphism A → I where I is an injective object in A.
Applying the functor F to this sequence, and chopping off the first term, we obtain the cochain complex Note: this is in general not an exact sequence anymore.
But we can compute its cohomology at the i-th spot (the kernel of the map from F(Ii) modulo the image of the map to F(Ii)); we call the result RiF(X).
Of course, various things have to be checked: the result does not depend on the given injective resolution of X, and any morphism X → Y naturally yields a morphism RiF(X) → RiF(Y), so that we indeed obtain a functor.
(Technically, to produce well-defined derivatives of F, we would have to fix an injective resolution for every object of A.
This choice of injective resolutions then yields functors RiF.
The above-mentioned property of turning short exact sequences into long exact sequences is a consequence of the snake lemma.
In practice, this fact, together with the long exact sequence property, is often used to compute the values of right derived functors.
An equivalent way to compute RiF(X) is the following: take an injective resolution of X as above, and let Ki be the image of the map Ii-1→Ii (for i=0, define Ii-1=0), which is the same as the kernel of Ii→Ii+1.
is a projective object), then one can define analogously the left-derived functors
to this sequence, chop off the last term, and compute homology to get
The short exact sequence is turned into the long exact sequence These left derived functors are zero on projectives and are therefore computed via projective resolutions.
; The category of modules has enough projectives so that left derived functors always exists.
This includes several notions of homology as special cases.
This gives rise to the derived tensor product
Derived functors and the long exact sequences are "natural" in several technical senses.
First, given a commutative diagram of the form (where the rows are exact), the two resulting long exact sequences are related by commuting squares:
Furthermore, this functor is compatible with the long exact sequences in the following sense: if is a short exact sequence, then a commutative diagram
In 1968 Quillen developed the theory of model categories, which give an abstract category-theoretic system of fibrations, cofibrations and weak equivalences.
Typically one is interested in the underlying homotopy category obtained by localizing against the weak equivalences.
For example, the category of topological spaces and the category of simplicial sets both admit Quillen model structures whose nerve and realization adjunction gives a Quillen adjunction that is in fact an equivalence of homotopy categories.
Particular objects in a model structure have “nice properties” (concerning the existence of lifts against particular morphisms), the “fibrant” and “cofibrant” objects, and every object is weakly equivalent to a fibrant-cofibrant “resolution.” Although originally developed to handle the category of topological spaces Quillen model structures appear in numerous places in mathematics; in particular the category of chain complexes from any Abelian category (modules, sheaves of modules on a topological space or scheme, etc.)
admit a model structure whose weak equivalences are those morphisms between chain complexes preserving homology.
Often we have a functor between two such model categories (e.g. the global sections functor sending a complex of Abelian sheaves to the obvious complex of Abelian groups) that preserves weak equivalences within the subcategory of “good” (fibrant or cofibrant) objects.
By first taking a fibrant or cofibrant resolution of an object and then applying that functor, we have successfully extended it to the whole category in such a way that weak equivalences are always preserved (and hence it descends to a functor from the homotopy category).
Applying these to a sheaf of Abelian groups interpreted in the obvious way as a complex concentrated in homology, they measure the failure of the global sections functor to preserve weak equivalences of such, its failure of “exactness.” General theory of model structures shows the uniqueness of this construction (that it does not depend of choice of fibrant or cofibrant resolution, etc.)