nLab
derived functor

Context

Homotopy theory

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model category

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Universal constructions

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Producing new model structures

Presentation of (,1)(\infty,1)-categories

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Contents

Idea

A derived functor is a functor in homotopy theory induced from, “derived from” or presented by an ordinary functor on a category with weak equivalences.

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.

General idea

A category with weak equivalences CC serves as a presentation for an (∞,1)-category C\mathbf{C} by simplicial localization. Accordingly, a functor F:CDF : C \to D should induce an (∞,1)-functor between the corresponding (∞,1)-categories CD\mathbf{C} \to \mathbf{D}. From the nPOV, this is a derived functor.

If FF is a homotopical functor in that it respects the weak equivalences in CC and DD, 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 x,yCx, y \in C that are equivalent in C\mathbf{C} but not isomorphic in CC, the functor will in general not assign objects F(x)F(x) and F(y)F(y) that are equivalent in D\mathbf{D}, as an (∞,1)-functor would. So it matters on which representatives of a C\mathbf{C}-equivalence class of objects the functor FF is applied.

Remembering that by Dwyer-Kan simplicial localization the morphisms in C\mathbf{C} and D\mathbf{D} are zig-zags of morphisms in CC and DD, a very general notion of derived functor therefore takes a derived functor of FF to be a functor 𝔻F:CD\mathbb{D}F : \mathbf{C} \to \mathbf{D} induced from the universal property of the localization by a functor of the form FQ:CDF \circ Q : C \to D, where Q:CCQ : C \to C is an endofunctor which is naturally connected to the identity by a zig-zag of weak equivalences:

XX 1X 2QX. X \stackrel{\simeq}{\leftarrow} X_1 \stackrel{\simeq}{\to} X_2 \cdots \stackrel{\simeq}{\leftarrow} Q X \,.

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 the presence of model category structure

In highly structured situations where CC and DD are equipped not just with weak equivalences but with the full structure of a model category and if FF 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 𝕃F:CD\mathbb{L}F : \mathbf{C} \to \mathbf{D} of a left Quillen functor F:CDF : C \to D is obtained by applying FF to cofibrant objects of CC. Similarly a right derived functor G:DC\mathbb{R}G : \mathbf{D} \to \mathbf{C} of a right Quillen functor G:DCG : D \to C is obtained by applying GG to fibrant objects.

Recalling that the (∞,1)-category presented by a simplicial model category CC may be identified with the full sSet-subcategory C C^\circ of fibrant-cofibrant objects, this may be understood as ensuring that the derived functor indeed respects the (,1)(\infty,1)-categorical structure. More precisely, for

(FG):CD (F \dashv G) : C \stackrel{\leftarrow}{\to} D

an sSet-enriched Quillen adjunction between simplicial model categories, combining FF and GG with cofibrant-fibrant replacement induces a pair of adjoint (∞,1)-functors

𝕃F:CD:G \mathbb{L}F : \mathbf{C} \stackrel{\leftarrow}{\to} \mathbf{D} : \mathbb{R}G

between quasi-categories C=N(C )\mathbf{C} = N(C^\circ), D=N(D )D = N(D^\circ), where NN is the homotopy coherent nerve functor.

Often a simplified version of this situation is considered, where instead of the (∞,1)-categories C\mathbf{C} and D\mathbf{D} only their homotopy categories are remembered, equivalently the homotopy categories of CC and DD. The above adjoint (∞,1)-functors restrict to functors

LF:Ho(C)Ho(D):RG L F : Ho(C) \stackrel{\leftarrow}{\to} Ho(D) : R G

on homotopy categories, and often its is these functors that are called (total) derived functors in the literature

More generally, derived functors in this sense may be considered in situations where less than the above extra structure is available (no model category structure or not Quillen adjunction).

On homotopy categories

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 f:CDf : C \to D is to extend it to a diagram

C F D Q C (?) Q D Ho(C) Ho(D), \array{ C &\stackrel{F}{\to}& D \\ \downarrow^{\mathrlap{Q_C}} &(?)& \downarrow^{\mathrlap{Q_D}} \\ Ho(C) &\to& Ho(D) } \,,

where Q C:CHo(C)Q_C : C \to Ho(C) is the universal morphism characterizing the homotopy category and similarly for Q DQ_D.

There is a general method of ordinary category theory to solve such problems universally: one may take Ho(C)Ho(D)Ho(C) \to Ho(D) to be either the left or right Kan extension of Q dFQ_d \circ F along Q CQ_C.

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 FF with cofibrant/fibrant or similar replacement functors. It then also happens that the functors so obtained are left or right Kan extensions.

Definition

We first give the decategorified definition of total derived functors on homotopy categories in

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.

As functors on homotopy categories

For Core(C)WCCore(C) \hookrightarrow W \hookrightarrow C a category with weak equivalences, then for F:CDF : C \to D any functor, the left derived functor LFL F of FF is the right Kan extension of FF along the projection p:CHo Cp : C \to Ho_C to the homotopy category

C F D p LF Ho C \array{ C &&\stackrel{F}{\to}&& D \\ \downarrow^p &\Downarrow& \nearrow_{L F} \\ Ho_C }

(if it exists). Dually, the right derived functor RFR F of FF is its left Kan extension along pp. Note the reversal of handedness; this is unfortunate but unavoidable.

More generally, if DD is itself a category with weak equivalences, then by derived functors of FF we often mean derived functors of the composite

CFDHo D C \stackrel{F}{\to} D \to Ho_D
Remark

By the universal property of Ho CHo_C, functors Ho CDHo_C \to D are equivalent to functors CDC\to D which take weak equivalences to isomorphisms. If FF itself takes weak equivalences to isomorphisms, then its left and right derived functors are both (isomorphic to) its unique extension along pp. In general, however, LFL F and RFR F are not extensions of FF even up to isomorphism.

Remark

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.

Remark

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.

As functors on (,1)(\infty,1)-categories

Proposition

Let CC and DD by simplicial model categories and let

(FG):CFGD (F \dashv G) : C \stackrel{\overset{G}{\leftarrow}}{\underset{F}{\to}} D

be an sSet-enriched Quillen adjunction. Then there is an (∞,1)-adjunction

(𝕃FG):N(C )𝕃FGN(D ) (\mathbb{L}F \dashv \mathbb{R}G) : N(C^\circ) \stackrel{\overset{\mathbb{R} G}{\leftarrow}}{\underset{\mathbb{L}F}{\to}} N(D^\circ)

between quasi-categories N(C )N(C^\circ) and N(D )N(D^\circ) 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

(LFRG):Ho(C)LFRGHo(D). (L F \dashv R G) : Ho(C) \stackrel{\overset{R G}{\leftarrow}}{\underset{L F}{\to}} Ho(D) \,.

This is prop. 5.2.4.6 and remark 5.2.4.7 in (Lurie).

In homological algebra

Often and traditionally, the concept of derived functors is considered in homological algebra exclusively in the context of categories of chain complexes Ch (𝒜)Ch_\bullet(\mathcal{A}) in an abelian category 𝒜\mathcal{A}.

By taking quasi-isomorphisms as weak equivalences, Ch (𝒜)Ch_\bullet(\mathcal{A}) 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 𝒜\mathcal{A} are required in order for these model structures to exist.)

Now, any ordinary additive functor F:𝒜F\colon \mathcal{A} \to \mathcal{B} between abelian categories induces a functor Ch (F):Ch (𝒜)Ch ()Ch_\bullet(F)\colon Ch_\bullet(\mathcal{A}) \to Ch_\bullet(\mathcal{B}) between categories of chain complexes. We can therefore ask about derived functors of Ch (F)Ch_\bullet(F).

Note first that Ch (F)Ch_\bullet(F) 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 Ch (F)Ch_\bullet(F) 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 Ch (𝒜)Ch_\bullet(\mathcal{A}) exists, and a right derived functor if the injective model structure exists.

In the homological algebra literature, what is called the ppth right derived functor

R pF:𝒜 R^p F \colon \mathcal{A} \to \mathcal{B}

is the composite

𝒜B p()Ch (𝒜)Ch (F)Ch ()H 0(), \mathcal{A} \stackrel{\mathbf{B}^p (-)}{\hookrightarrow} Ch_\bullet(\mathcal{A}) \stackrel{\mathbb{R} Ch_\bullet(F)}{\to} Ch_\bullet(\mathcal{B}) \stackrel{H^0(-)}{\to} \mathcal{B} \,,
  1. The first map sends an object A𝒜A \in \mathcal{A} to the corresponding Eilenberg-MacLane object B pA\mathbf{B}^p A: the cochain complex A[p]A[p] concentrated on AA in degree pp.

  2. The second map is the actual right derived functor Ch (F)\mathbb{R}Ch_\bullet(F) of Ch (F)Ch_\bullet(F) in the sense used previously on this page. Thus, this is itself the composite

    F:Ch (𝒜)PCh (𝒜)Ch (F)Ch (), \mathbb{R}F : Ch_\bullet(\mathcal{A}) \stackrel{P}{\to} Ch_\bullet(\mathcal{A}) \stackrel{Ch_\bullet(F)}{\to} Ch_\bullet(\mathcal{B}) \,,

    where PP 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.

  3. The last morphism computes the cochain cohomology of the resulting cochain complex in degree 0.

Of course, it is equivalent to instead regard AA as concentrated in degree 00, and then take the ppth 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:

  1. Instead of applying the first step and restricting attention to arguments that are chain complexes concentrated in a single degree, one can evaluate Ch (F)\mathbb{R} Ch_\bullet(F) on all chain complexes (and then, if desired, take homology groups). In homological algebra one then speaks of hyper-derived functors.

  2. 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.

Long exact sequences

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

0ABC0 0 \to A \to B \to C \to 0

in 𝒜\mathcal{A}. Assuming 𝒜\mathcal{A} has enough projectives, we can then find projective resolutions QAQ A, QBQ B, and QCQ C of AA, BB, and CC, respectively, such that

0QAQBQC0 0 \to Q A \to Q B \to Q C \to 0

is a short exact sequence of chain complexes. But since QCQ C is projective, this short exact sequence is split, and therefore preserved by any additive functor. Thus we have another short exact sequence

FQAFQBFQC F Q A \to F Q B \to F Q C

which therefore gives rise to a long exact sequence in homology:

H 1(FQA)H 1(FQB)H 1(FQC)H 0(FQA)H 0(FQB)H 0(FQC). \cdots \to H_1(F Q A) \to H_1(F Q B) \to H_1(F Q C) \to H_0 (F Q A) \to H_0(F Q B) \to H_0(F Q C).

Of course, these homology groups are precisely the left derived functors of FF, in the traditional homological algebra sense, applied to AA, BB, and CC.

All of this works without hypothesis on FF. However, if FF is right exact, then it preserves the exactness of the sequence

QA 1QA 0A0 Q A_1 \to Q A_0 \to A \to 0

(and the analogous ones for BB and CC. This implies that FAH 0(FQA)F A \cong H_0 (F Q A) and so on, so that the above long exact sequence actually finishes

H 1(FQA)H 1(FQB)H 1(FQC)FAFBFC0. \cdots \to H_1(F Q A) \to H_1(F Q B) \to H_1(F Q C) \to F A \to F B \to F C \to 0.

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.

Examples

References

General

General discussion of derived functors in homotopy theory is for instance in

Discussion in the context of (∞,1)-categories is in section 5.2.4 of

In homological algebra

An standard textbook introduction to derived functors in homological algebra is in

A systematic discussion of this case from the point of view of localization and homotopy theory is in section 13 of

and, similarly, in section 7 of

Revised on October 13, 2013 02:44:33 by Urs Schreiber (82.113.121.239)