Гомологиялық құрылымдар және турастырылған функторлар

Homological construction in category theory

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 : A → B between two abelian categories A and B. If 0 → A → B → C → 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: A → B, 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.

If is a ring, then the category of all left modules is an abelian category with enough injectives. If is a fixed left module, then the functor is left exact, and its right derived functors are the Ext functors Alternatively can also be obtained as the left derived functor of the right exact functor
Various notions of cohomology are special cases of Ext functors and therefore also derived functors. Group cohomology is the right derived functor of the invariants functor which is the same as (where is the trivial module) and therefore Lie algebra cohomology of a Lie algebra over some commutative ring is the right derived functor of the invariants functor which is the same as (where is again the trivial module and is the universal enveloping algebra of ). Therefore Hochschild cohomology of some algebra is the right derived functor of invariants mapping a bimodule to its center, also called its set of invariants which is the same as (where is the enveloping algebra of and is considered an bimodule via the usual left and right multiplication). Therefore :

The category of left modules also has enough projectives. If is a fixed right module, then the tensor product with gives a right exact covariant functor ; The category of modules has enough projectives so that left derived functors always exists. The left derived functors of the tensor functor are the Tor functors Equivalently can be defined symmetrically as the left derived functors of In fact one can combine both definitions and define as the left derived of
This includes several notions of homology as special cases. This often mirrors the situation with Ext functors and cohomology. Group homology is the left derived functor of taking coinvariants which is the same as Lie algebra homology is the left derived functor of taking coinvariants which is the same as Hochschild homology is the left derived functor of taking coinvariants which is the same as
Instead of taking individual left derived functors one can also take the total derived functor of the tensor functor. This gives rise to the derived tensor product where is the derived category.

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.)