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Let $\mathcal A$ be an abelian category. In this lecture, Thomas Nikolaus

  1. Defines the unbounded derived category $\mathcal D(\mathcal A)$ as $\mathcal K(\mathcal A)[W^{-1}]$, where $\mathcal K(\mathcal A)=N_{\mathrm{dg}}(\operatorname{Ch}(\mathcal A))$ and $W$ is the set of quasi-isomorphisms of chain complexes; and
  2. Defines the notion of left-derived functor from an $\infty$-category with a set of weak equivalences to an arbitrary $\infty$-category, and shows that the left derived functor of a functor from $\mathcal K(\mathcal A)$ to an arbitrary $\infty$-category can be computed as a limit, provided it exists.

Point (1) is identical to the construction of the unbounded derived category of an abelian category as a triangulated category, before you know anything about K-injective complexes (Lurie’s approach in Higher Algebra is to require the existence of K-injectives to define the $\infty$-categorical unbounded derived category). Point (2) is identical to Deligne’s approach to defining derived functors (05S7).

My question is, where is (2), or more generally an $\infty$-categorical notion of derived functor, written down? I wasn’t even able to find in HTT, HA, or SAG a definition of derived functor from an $\infty$-category with weak equivalences to another $\infty$-category, but I may have overlooked it in the thousands of pages. I’m especially interested in Nikolaus’s computation of derived functors out of $\mathcal K(\mathcal A)$ as a limit/colimit because it’s easy to connect to everything that’s written down in the language of triangulated categories (e.g. you immediately know you can compute the derived tensor product using a K-flat resolution etc.).

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2 Answers 2

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An account of derived functors between ∞-categories equipped with weak equivalences and fibrations can be found in Section 7.5 of Cisinski's Higher Categories and Homotopical Algebra. This setting is sufficient to treat the case of the unbounded derived category of an abelian category.

A more general treatment of derived functors in the setting of ∞-categories equipped with a calculus of fractions can be found in Section 7.2. See, in particular, Theorem 7.2.8 and Corollary 7.2.9, as well as Remark 7.2.21, which connects this setting to the setting of fibrations.

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  • $\begingroup$ While this is a great reference, I don't think it covers the general notion of derived functors (it always assumes that the source is an $\infty$-category with weak equivalences and fibrations), and in particular it does not prove the formula asked by the OP (although it does prove it in this special case, where it is substantially simpler) $\endgroup$ Jul 12 at 9:39
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    $\begingroup$ @DenisNardin I do define in my book derived functors as Kan extensions along the localization of any marked ∞-category. The general theory of derived functor simply is the general theory of Kan extensions. The case of ∞-categories with weak equivalences and fibrations covers at the very least cochain complexes in an abelian category, and the point is to construct these Kan extensions using resolutions. Deligne's formula is a variation on the fact that such derived functors are pointwise, so that we can understand them after embedding the codomain in a suitable presheaf category. $\endgroup$ Jul 12 at 12:26
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    $\begingroup$ To be more precise: Deligne's formula is essentially a combination of Corollary 7.2.9 and Theorem 7.3.16 in my book (in the case of Verdier quotients, we have a filtered colimit, so that this formula is even nicer). Paragraph 7.5.25 constructs derived functors using resolutions and compares with Deligne's formula (in order to prove the expected universal property). $\endgroup$ Jul 12 at 12:32
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    $\begingroup$ @DenisNardin In fact Corollary 7.5.17 says that, in general, in the case of ∞-category with weak equivalences and fibrations, the right derived functor à la Deligne always coincides with the right derived functors of the restriction to fibrant objects, and Lemma 7.5.24 tells us that a functor which sends weak equivalences to isomorphisms has a trivial derived functor. This explains why we have derived functors using resolutions (7.5.25). Derived functors defined using resolutions are in fact absolute (7.5.25.3) and this explains why we have derived adjunctions (Theorem 7.5.30). $\endgroup$ Jul 12 at 19:59
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    $\begingroup$ In the comment above, the reference to 7.3.16 is typo: should be 7.2.16 $\endgroup$ Jul 13 at 10:04
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I’d like to expand on the accepted answer and comments.

Cisinski describes in §7.5.25 of his book how to compute right derived functors in the context of an $\infty$-category $C$ with weak equivalences and fibrations. This is good enough to compute $RF$ if $F:K(A):=N_{\mathrm{dg}}(\operatorname{Ch}A)\to D$ with $A$ a Grothendieck abelian category and $D$ any $\infty$-category: one takes quasi-isomorphisms as the weak equivalences in $K(A)$ and the fibrations to be as in the model structure on $\operatorname{Ch}A$ of HA.1.3.5.3. That this satisfies the axioms of Cisinski’s Definition 7.4.12 is easy to check, since a square in $K(A)$ is cartesian iff it’s cocartesian iff the map on cones is an equivalence in $K(A)$; i.e. a chain homotopy equivalence (HA.1.2.4.14 & HA.1.3.2.17). If $D=D(B):=K(B)[\mathrm{q.i.}^{-1}]$ with $B$ an abelian category then this computes $RF$ for any additive functor $F:A\to B$.

What about tensor product? Well, in that case one would like to find cofibrations that together with the weak equivalences make $K(A)$ into an $\infty$-category with weak equivalences and cofibrations and so that the cofibrant objects are the K-flat complexes. I don’t know how to do this in general, but when $A=\operatorname{Mod}_R$ then a preprint of Gillespie shows there exists a model structure on $\operatorname{Ch}A$ whose cofibrant objects are the K-flat complexes. In any case, it seems at least superficially easier to compute with Deligne’s formula than to find the right fibrations or cofibrations, so let’s turn to that.

Proposition Let $C$ and $W$ be as in §7.2.1 of Cisinski’s book and let $D$ be a (co)complete $\infty$-category. Suppose given a right (left) calculus of fractions $W(x)$ at $x\in C$ and let $\gamma:C\to C[W^{-1}]$ be the localization map. Then $$(\gamma_*F)(\gamma x)\simeq\lim_{z_0\to x_0}F(z),$$ where the limit is indexed over $W(x)$ (respectively $$(\gamma_!F)(\gamma x)\simeq\operatorname{colim}_{x_0\to z_0}F(z),$$ indexed over $W(x)$).

(Here, $\gamma_*:\operatorname{Fun}(C,D)\to\operatorname{Fun}(C[W^{-1}],D)$ is the functor of right Kan extension, so $\gamma_*F$ computes $LF$, while $\gamma_!:\operatorname{Fun}(C,D)\to\operatorname{Fun}(C[W^{-1}],D)$ is left Kan extension and $\gamma_!F=RF$.)

Corollary Let $A$ be an abelian category, $D$ a cocomplete $\infty$-category, and $F:K(A)\to D$ a functor. If $K\in K(A)$, $$RF(K)\simeq\operatorname{colim}_{K\to K’} F(K’),$$ where the (filtered) colimit is indexed by $K(A)_{K/}^{\mathrm{q.i.}}$, the full subcategory of $K(A)_{K/}$ on the quasi-isomorphisms. Dually, if $D$ is instead complete, $$LF(K)\simeq\lim_{K’\to K}F(K’),$$ where the (cofiltered) limit is indexed by $K(A)_{/K}^{\mathrm{q.i.}}$.

The corollary follows from the proposition using Cisinski’s Theorem 7.2.16, as the set of quasi-isomorphisms is closed under composition as well as pullback and pushout in $K(A)$.

Proof of Proposition – We can rewrite (the dual of) Cisinski’s Corollary 7.29 as follows: if $F$ is a functor $C\to\mathcal S(=\mathrm{Kan})$ and there is a right calculus of fractions $W(x)$ at $x\in C$, then $$\lim_{z_0\to x_0}F(z)\simeq(\gamma_*F)(\gamma x),$$ where the limit is indexed over $W(x)$. If $G:X\to Y$ is any functor of simplicial sets and $x\in X$, let $G_x:=x^*G$ denote the $x$-fiber of $G$. The functor $$\gamma_*:=\operatorname{Fun}(X,\gamma_*):\operatorname{Fun}(X,\operatorname{Fun}(C,\mathcal S))\to\operatorname{Fun}(X,\operatorname{Fun}(C[W^{-1}],\mathcal S))$$ is functorial in $X$, so in particular $\gamma_*(\Phi)_x=\gamma_*\Phi_x$ for any $x\in X$ and functor $\Phi:X\to\operatorname{Fun}(C,\mathcal S)$. Letting $X=D^{\mathrm{op}}$ and $\Phi=h_DF=\operatorname{Map}_D(-,F):C\to\operatorname{Fun}(D^{\mathrm{op}},\mathcal S)$, we produce via right Kan extension a functor $\gamma_*(h_DF):C[W^{-1}]\to\operatorname{Fun}(D^{\mathrm{op}},\mathcal S)$ so that for each $d\in D$ and $x\in C$, $$(\gamma_*(h_DF)_{\gamma x})_d=\gamma_*\operatorname{Map}_D(d,F)_{\gamma x}=\lim_{z_0\to x_0}\operatorname{Map}_D(d,Fz)=\operatorname{Map}_D(d,\lim_{z_0\to x_0}F(z)).$$ On the other hand, the functor $$\alpha:\operatorname{Fun}(C[W^{-1}],h_D):\operatorname{Fun}(C[W^{-1}],D)\to\operatorname{Fun}(C[W^{-1}],\operatorname{Fun}(D^{\mathrm{op}},\mathcal S))$$ sends $\gamma_*F=LF$ to $\gamma_*(h_DF)$ by Cisinski’s Proposition 6.4.9. Therefore $$(\gamma_*(h_DF)_{\gamma x})_d=(\alpha(LF)_{\gamma x})_d=\operatorname{Map}_D(-,LF(\gamma x))_d=\operatorname{Map}_D(d,LF(\gamma x)).\qquad\square$$

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    $\begingroup$ We may consider several structures of categories with weak equivalences and fibrations on $K(A)$ (itself the localization of the $1$-category of cochain complexes by cochain homotopy equivalences, which is a model category if one takes degreewise split mono's as cofibrations): the weak equivalences are quasi-isomorphisms, and the fibrations are all maps (we can also take cofibrations to be all maps). The corresponding calculi of fractions is then the one you describe in the second part of your post. $\endgroup$ Jul 19 at 9:29
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    $\begingroup$ If you have enough flat objects in $A$ (i.e. for any object $a$ in $A$, there is an epimorphism $x\to a$ with $x$ flat (in the sense the $x\otimes(-)$ is exact), then we can consider a class of cofibrations in $K(A)$: the maps with cofiber isomorphic in $K(A)$ (i.e. cochain homotopy equivalent) to a complex of flat objects in $A$ (no need of complicated conditons such as $K$-flatness: these are only there to get Quillen model structures and we do not care about that here). $\endgroup$ Jul 19 at 9:33
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    $\begingroup$ The left calculus of fractions induced by these cofibrations is what allows you to prove that the derived tensor product exists. If $A$ is Grothendieck, you can find cofinal functors with small domain with values in the indexing categories, which is why derived functors exist whenever the target has small (co)limits. Otherwise, everything we say above works for any small abelian category, if we consider derived functors with values in a (co)complete $\infty$-category. $\endgroup$ Jul 19 at 9:35
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    $\begingroup$ The smallness up to cofinality comes from the fact that, in the 1-category of cochain complexes, the subcategory of quasi-isomorphisms is accessible (being the inversible Image of the accessible subcategory of isomorphisms through the small family of accessible functors $H^n$). It is not difficult but writing it down property in a convincing manner requires some effort. $\endgroup$ Jul 20 at 6:56
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    $\begingroup$ There is a similar story with flat resolutions. The proofs that we can express all this via suitable model structures usually also give you the tools to deal with size issues (the fact that we can apply the small object argument is directly related to that). $\endgroup$ Jul 20 at 7:00

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