related by the Dold-Kan correspondence
Historically the concept first arose in the special context of homological algebra on categories of chain complexes and is often still understood by default in this special sense. The relation to the general case is discussed below in the section In homological algebra. For a dedicated discussion of this case see the entry derived functor in homological algebra.
A category with weak equivalences serves as a presentation for an (∞,1)-category by simplicial localization. Accordingly, a functor should induce an (∞,1)-functor between the corresponding (∞,1)-categories . From the nPOV, this is a derived functor.
If is a homotopical functor in that it respects the weak equivalences in and , then by the universal property of simplicial localization it extends to a functor of (∞,1)-categories and this is the corresponding derived functor .
However, typically functors of interest do not respect weak equivalences and hence do not uniquely or even naturally give rise to an (∞,1)-functor. In general, they contain too little information to accomplish this. Notably, to objects that are equivalent in but not isomorphic in , the functor will in general not assign objects and that are equivalent in , as an (∞,1)-functor would. So it matters on which representatives of a -equivalence class of objects the functor is applied.
Remembering that by Dwyer-Kan simplicial localization the morphisms in and are zig-zags of morphisms in and , a very general notion of derived functor therefore takes a derived functor of to be a functor induced from the universal property of the localization by a functor of the form , where is an endofunctor which is naturally connected to the identity by a zig-zag of weak equivalences:
Here if this zig-zag consists just of one morphism to the left one would speak of a left derived functor. If it consistis of just one morphism to the right, one would speak of a right derived functor. In general, it is just a derived functor.
In highly structured situations where and are equipped not just with weak equivalences but with the full structure of a model category and if is a left or right Quillen functor with respect to these model structures, there are accordingly more structured ways to solve this problem:
the left derived functor of a left Quillen functor is obtained by applying to cofibrant objects of . Similarly a right derived functor of a right Quillen functor is obtained by applying to fibrant objects.
Recalling that the (∞,1)-category presented by a simplicial model category may be identified with the full sSet-subcategory of fibrant-cofibrant objects, this may be understood as ensuring that the derived functor indeed respects the -categorical structure. More precisely, for
Often a simplified version of this situation is considered, where instead of the (∞,1)-categories and only their homotopy categories are remembered, equivalently the homotopy categories of and . The above adjoint (∞,1)-functors restrict to functors
If one forgets the nPOV and that a category with weak equivalences should be regarded as presentation for an (∞,1)-category, then it might seem as if all one wants when deriving a homotopical functor is to extend it to a diagram
where is the universal morphism characterizing the homotopy category and similarly for .
This is often in the literature given as the definition of, respectively , total left and right derived functors. Unfortunately, it is not clear how this definition by Kan extension relates to what should be the right (∞,1)-category theory picture. Moreover, the examples of derived functors that play any practical role are effectively always constructed instead rather by combining with cofibrant/fibrant or similar replacement functors. It then also happens that the functors so obtained are left or right Kan extensions.
and then the (∞,1)-category-version in
The special case of derived functors in the context of homological algebra is discussed from this general perspective in
A dedicated discussion of this case is at derived functors in homological algebra.
(if it exists). Dually, the right derived functor of is its left Kan extension along . Note the reversal of handedness; this is unfortunate but unavoidable.
More generally, if is itself a category with weak equivalences, then by derived functors of we often mean derived functors of the composite
By the universal property of , functors are equivalent to functors which take weak equivalences to isomorphisms. If itself takes weak equivalences to isomorphisms, then its left and right derived functors are both (isomorphic to) its unique extension along . In general, however, and are not extensions of even up to isomorphism.
In practice, derived functors are usually computed using fibrant and cofibrant resolution replacements (see the entries on homotopy theory and model category) or, more generally, deformation retracts.
If the codomain admits sufficiently many limits and colimits, a Kan extension can be computed in terms of those, and that such Kan extensions are called pointwise. Homotopy categories generally do not admit even small limits and colimits, and moreover the domains of the functors in question are generally large, so such a construction of a derived functor is not possible.
However, when derived functors are constructed using fibrant and cofibrant replacements, as above, it turns out a posteriori that they are actually pointwise: they are preserved by all representable functors, and hence their individual object values have the universal property of the (generally large) limits that would have been used to compute them, even though not all limits exist in the homotopy category. In fact, derived functors constructed in this way are actually absolute Kan extensions: preserved by any functor whatsoever.
Let and by simplicial model categories and let
between quasi-categories and otained as the homotopy coherent nerves of the full sSet-subcategories of fibrant-cofibrant objects. Their image on the homotopy categories produces the notion of total derived functor between homotopy categories discussed above
This is prop. 18.104.22.168 and remark 22.214.171.124 in (Lurie).
By taking quasi-isomorphisms as weak equivalences, is naturally a category with weak equivalences. In much of the literature on homological algebra, the refinement of this structure to a projective or injective model structure on chain complexes is implicit. For instance, an injective resolution of chain complexes is nothing but a fibrant replacement in the injective model structure. Dually, a projective resolution is a cofibrant replacement in the projective model structure. (Note, though, that hypotheses on are required in order for these model structures to exist.)
Note first that automatically preserves chain homotopies, and therefore also preserves chain homotopy equivalence?s. Since the projective (resp. injective) model structure on chain complexes has the property that weak equivalences (that is, quasi-isomorphisms) between cofibrant (resp. fibrant) objects are chain homotopy equivalences, it follows that automatically preserves weak equivalences between projective-cofibrant objects, and also between injective-fibrant objects. Thus, it has a left derived functor if the projective model structure on exists, and a right derived functor if the injective model structure exists.
In the homological algebra literature, what is called the th right derived functor
is the composite
The first map sends an object to the corresponding Eilenberg-MacLane object : the cochain complex concentrated on in degree .
The second map is the actual right derived functor of in the sense used previously on this page. Thus, this is itself the composite
where denotes a fibrant resolution functor in the injective model structure on chain complexes. Applied to an Eilenberg-MacLane object, this amounts to the usual injective resolutions seen in the homological algebra literature.
The last morphism computes the cochain cohomology of the resulting cochain complex in degree 0.
Of course, it is equivalent to instead regard as concentrated in degree , and then take the th homology group at the last step. Left derived functors are dual, using the projective model structure.
The first and the last steps are traditionally included, but are not really necessary:
Instead of applying the first step and restricting attention to arguments that are chain complexes concentrated in a single degree, one can evaluate on all chain complexes (and then, if desired, take homology groups). In homological algebra one then speaks of hyper-derived functors.
The last step of taking cohomology groups serves to extract invariant and computable information. It also destroys the simple composition law of functors, though. But there is a computational tool that can be used to recover the derived functor – in this homological sense – of the composite of two functors from their individual derivations: this is the spectral sequence called the Grothendieck spectral sequence.
Traditionally, in homological algebra, one only takes left derived functors of right exact functors, and right derived functors of left exact ones. As we saw above, both left and right derived functors can be defined without these hypotheses, but it is only in the presence of these hypotheses that we obtain long exact sequences.
Specifically, suppose we have a short exact sequence
in . Assuming has enough projectives, we can then find projective resolutions , , and of , , and , respectively, such that
is a short exact sequence of chain complexes. But since is projective, this short exact sequence is split, and therefore preserved by any additive functor. Thus we have another short exact sequence
which therefore gives rise to a long exact sequence in homology:
Of course, these homology groups are precisely the left derived functors of , in the traditional homological algebra sense, applied to , , and .
All of this works without hypothesis on . However, if is right exact, then it preserves the exactness of the sequence
(and the analogous ones for and . This implies that and so on, so that the above long exact sequence actually finishes
This is how derived functors are traditionally introduced in homological algebra: as a way to continue the right half of a short exact sequence preserved by a right exact functor into a long exact sequence. The case of left exact functors and right derived functors is dual.
In the context of a model structure on chain complexes of modules the left and right derived functors of the tensor product functor and the hom-functor are called Tor-functor and Ext-functor, respectively.
General discussion of derived functors in homotopy theory is for instance in
William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories , volume 113 of Mathematical Surveys and Monographs
Discussion in the context of (∞,1)-categories is in section 5.2.4 of
An standard textbook introduction to derived functors in homological algebra is in
and, similarly, in section 7 of