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

Question

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?

Answer 1

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 $$ \pi_* LG(\pi_* \operatorname{Ani}(F)(X)) \Rightarrow \pi_*(G\circ F)(X) $$