Derived functor

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.

It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations.

Suppose we are given a covariant left exact functor F : AB between two abelian categories A and B. If 0 → ABC → 0 is a short exact sequence in A, then applying F yields the exact sequence 0 → F(A) → F(B) → F(C) and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) there is one canonical way of doing so, given by the right derived functors of F. For every i≥1, there is a functor RiF: AB, and the above sequence continues like so: 0 → F(A) → F(B) → F(C) → R1F(A) → R1F(B) → R1F(C) → R2F(A) → R2F(B) → ... . From this we see that F is an exact functor if and only if R1F = 0; so in a sense the right derived functors of F measure "how far" F is from being exact.

If the object A in the above short exact sequence is injective, then the sequence splits. Applying any additive functor to a split sequence results in a split sequence, so in particular R1F(A) = 0. Right derived functors (for i>0) are zero on injectives: this is the motivation for the construction given below.

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 AI where I is an injective object in A.

The right derived functors of the covariant left-exact functor F : AB are then defined as follows. Start with an object X of A. Because there are enough injectives, we can construct a long exact sequence of the form

where the I i are all injective (this is known as an injective resolution of X). Applying the functor F to this sequence, and chopping off the first term, we obtain the chain 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 end result does not depend on the given injective resolution of X, and any morphism XY naturally yields a morphism RiF(X) → RiF(Y), so that we indeed obtain a functor. Note that left exactness means that 0 →F(X) → F(I0) → F(I1) is exact, so R0F(X) = F(X), so we only get something interesting for i>0.

(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. Different choices of resolutions yield naturally isomorphic functors, so in the end the choice doesn't really matter.)

The above-mentioned property of turning short exact sequences into long exact sequences is a consequence of the snake lemma. This tells us that the collection of derived functors is a δ-functor.

If X is itself injective, then we can choose the injective resolution 0 → XX → 0, and we obtain that RiF(X) = 0 for all i ≥ 1. 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-1Ii (for i=0, define Ii-1=0), which is the same as the kernel of IiIi+1. Let φi : Ii-1Ki be the corresponding surjective map. Then RiF(X) is the cokernel of Fi).

If one starts with a covariant right-exact functor G, and the category A has enough projectives (i.e. for every object A of A there exists an epimorphism PA where P is a projective object), then one can define analogously the left-derived functors LiG. For an object X of A we first construct a projective resolution of the form

where the Pi are projective. We apply G to this sequence, chop off the last term, and compute homology to get LiG(X). As before, L0G(X) = G(X).

In this case, the long exact sequence will grow "to the left" rather than to the right:

One may also start with a contravariant left-exact functor F; the resulting right-derived functors are then also contravariant. The short exact sequence

These left derived functors are zero on projectives and are therefore computed via projective resolutions.

There are various notions of cohomology which are a special case of this:

Various notions of cohomology are special cases of Ext functors and therefore also derived functors.

This includes several notions of homology as special cases. This often mirrors the situation with Ext functors and cohomology.

Derived functors and the long exact sequences are "natural" in several technical senses.

(where the rows are exact), the two resulting long exact sequences are related by commuting squares:

Second, suppose η : FG is a natural transformation from the left exact functor F to the left exact functor G. Then natural transformations Riη : RiFRiG are induced, and indeed Ri becomes a functor from the functor category of all left exact functors from A to B to the full functor category of all functors from A to B. Furthermore, this functor is compatible with the long exact sequences in the following sense: if

Both of these naturalities follow from the naturality of the sequence provided by the snake lemma.

Conversely, the following characterization of derived functors holds: given a family of functors Ri: AB, satisfying the above, i.e. mapping short exact sequences to long exact sequences, such that for every injective object I of A, Ri(I)=0 for every positive i, then these functors are the right derived functors of R0.

The more modern (and more general) approach to derived functors uses the language of derived categories.

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. A Quillen adjunction is an adjunction between model categories that descends to an adjunction between the homotopy categories. 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). This is the “derived functor.” The “derived functors” of sheaf cohomology, for example, are the homologies of the output of this derived functor. 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.)