# In the not necessarily abelian cat setting, is there a Grothendieck spectral sequence for computing the homotopy of a composition of derived functors?

Recall the Grothendieck spectral sequence which computes the homology groups of a composition of left derived functors $$F$$ and $$G$$ on abelian categories: \begin{align*} E_{p,q}^{2}(A)=L_{p}G\circ L_{q}F(A)\Rightarrow L_{p+q}(G\circ F)(A) \end{align*} (note the nLab link is written cohomologically instead).

To consider derived functors on a not necessarily abelian category $$\mathcal{C}$$, we will adopt the language of animation as described in this paper due to Česnavičius and Scholze - in particular section 5. Namely, the animation of a category $$\mathcal{C}$$ is an $$\infty$$-category $$\mathrm{Ani}(\mathcal{C})$$ freely generated by $$\mathcal{C}^{sfp}$$, the strongly of finite presentation objects of $$\mathcal{C}$$, and a functor $$F:\mathcal{C}\to\mathcal{D}$$ which preserves sifted colimits has a unique animation $$\mathrm{Ani}(F):\mathrm{Ani}(\mathcal{C})\to\mathrm{Ani}(\mathcal{D})$$ which is computed on $$\mathcal{C}^{sfp}$$ and defined via Kan extension.

For example, for a fixed ring $$k$$, the cotangent complex $$L_{-/k}$$ is the animation of the functor of Kahler differentials $$\Omega_{-/k}$$ on the nonabelian category of $$k$$-algebras, and $$L_{A/k}$$ is explicitly computed by taking a polynomial simplicial resolution (or more generally a cofibrant replacement) $$P\to A$$ and then $$L_{A/k}:=\Omega_{P/k}\otimes_{P}A$$.

And in the abelian setting, animation is the derived functor; namely, given $$F:\mathcal{C}\to\mathcal{D}$$ which is right exact, a cofibrant replacement is a resolution by projective modules and $$\mathrm{Ani}(F)$$ is the functor $$LF:D(\mathcal{C})\to D(\mathcal{D})$$. For instance, $$\mathrm{Ani}(-\otimes_{k}M)=-\otimes_{k}^{L}M$$.

The Grothendieck spectral sequence operates under the following hypotheses (Theorems 2.1 and 2.2 in the above nLab link): first, under certain "nice" behavior of $$F$$ and $$G$$, the composition of derived functors $$LF\circ LG$$ is the derived functor of the composition $$L(F\circ G)$$, and second, if $$F$$ takes projective objects (or more generally $$F$$-acyclic objects) to $$G$$-acyclic objects, then the Grothendieck spectral sequence exists, page two is defined as above, and abuts as above.

Now let $$F$$ and $$G$$ be functors on not necessarily abelian categories. In the Česnavičius-Scholze paper linked above, at Proposition 5.1.5, one has similar "nice" behavior of $$F$$ and $$G$$ which will imply that the composition of derived functors $$\mathrm{Ani}(F)\circ\mathrm{Ani}(G)$$ is the derived functor of the composition $$\mathrm{Ani}(F\circ G)$$.

Here is my question: suppose we have the aforementioned "niceness" and also $$F:\mathcal{C}\to\mathcal{D}$$ takes $$\mathcal{C}^{sfp}$$ to $$\mathcal{D}^{sfp}$$ (my best understanding for an analog of the second hypothesis). Is it the case that we get a Grothendieck-style spectral sequence where \begin{align*} E_{p,q}^{2}(A)=\pi_{p}\mathrm{Ani}(G)\circ\pi_{q}\mathrm{Ani}(F)(A)\Rightarrow\pi_{p+q}\mathrm{Ani}(G\circ F)(A)? \end{align*}

It seems to me that the proof of the spectral sequence in the abelian setting goes through. If not, what is the obstruction?

Not in general, no. The problem is that animated functors play well with colimits, and homotopy groups play better with limits. However, if your functors $$\mathcal{A}\xrightarrow{F}\mathcal{B}\xrightarrow{G}\mathcal{C}$$ are such that $$G$$ is a right-exact functor of abelian categories, you do get a Grothendieck spectral sequence. In that case, $$\operatorname{Ani}(\mathcal{B})\subseteq \mathcal{D}(\mathcal{B})$$ is the full subcategory on connective objects, and $$\mathcal{D}(\mathcal{B})$$ is stable, same for $$\mathcal{C}$$. The animated functor $$\operatorname{Ani}(G\circ F)$$ can be recovered from the composite $$LG \circ \operatorname{Ani}(F): \operatorname{Ani}(\mathcal{A})\to \mathcal{D}(\mathcal{C})$$. For any $$X\in \operatorname{Ani}(\mathcal{A})$$, the Whitehead tower $$\tau_{\geq *} \operatorname{Ani}(F)(X)$$ gives a filtration on $$\operatorname{Ani}(F)(X)$$, which is taken by the exact $$LG$$ to a filtration on $$\operatorname{Ani}(G\circ F)(X)$$ with associated graded $$LG(\pi_* \operatorname{Ani}(F)(X))[*]$$. This gives a spectral sequence $$\pi_* LG(\pi_* \operatorname{Ani}(F)(X)) \Rightarrow \pi_*(G\circ F)(X)$$

• I heard that there are "unstable" spectral sequences (e.g. unstable Adams), and I saw a paper which states a spectral sequence for $\infty$-categories but I don't know whether it is related, due to my ignorance: arxiv.org/abs/2011.14931
– Z. M
Mar 6 at 14:28
• Thanks @Achim! What can be said when $G$ is not a right exact functor on abelian categories? Is that requirement strictly necessary, or are there other hypotheses on $G$, $\mathcal{B}$, and $\mathcal{C}$ that get us the spectral sequence? For instance, just to calibrate: take the setup in your answer; if $G$ is the Kahler differentials which takes a $k$-algebra $A$ to an $A$-module, would $Ani(G)=L_{-/k}$ enjoy the spectral sequence with $E_{p,q}^{2}=\pi_{p}L_{-/k}\circ\pi_{q}Ani(F)(X)\Rightarrow\pi_{p+q}Ani(\Omega_{-/k}\circ F)(X)$?
– Eric
Mar 6 at 22:45
• Oh, also if you get a chance in addition to the first question: do you have sources and further reading I can check out to familiarize myself?
– Eric
Mar 6 at 22:47
• For an animated ring, the homotopy groups are not rings, so the version you're suggesting doesn't even typecheck. I don't think you can relax the conditions on the second functor very much from what I wrote. Mar 7 at 1:14
• As for references, I'm sorry, but I can't really pin down a single source where I learned this stuff. Mar 7 at 1:17