Computing Ext groups in a functor stable $\infty$-category

Question

Let $I$ be a small category and $\mathcal{D}=D^b_\infty(\mathbb{Z})$ the bounded derived $\infty$-category of abelian groups. Consider the $\infty$-category $\mathcal{C}:=\mathrm{Fun}(I,\mathcal{D})$. Define a bounded t-structure on $\mathcal{C}$ by lifting the one on $\mathcal{D}$, that is $\mathcal{C}^{\leq 0}=\mathrm{Fun}(I,\mathcal{D}^{\leq 0})$. This is well defined because mapping spaces in $\mathcal{D}$ are computed as an end : if $F\in \mathcal{D}^{\leq 0}$ and $G\in \mathcal{D}^{\geq 1}$ then we have $\mathrm{Map}(F(i),G(j))=0$ for all $i,j\in I$ hence the bifunctor $\mathrm{Map}(F(-),G(=))$ is trivial and its end must be too. The heart of this t-structure is equivalent to the nerve of the abelian category of functors $I\to \mathbb{Z}\mathrm{-Mod}$. I am interested in computing $$ \mathrm{Ext}^i_{\mathcal{D}}(F,G):=\pi_0 \mathrm{Map}_{\mathcal{D}}(F,G[i]) $$ for ordinary functors $F,G:I\to \mathbb{Z}\mathrm{-Mod}$. This seems similar to the situation of the computation of Ext groups between abelian groups seen as objects in the stable infinity category of spectra (which seems to be something quite standard ; note though that I know very little algebraic topology), so I was wondering if it has already been treated somewhere or if some methods would translate.

We can wonder wether $\mathcal{C}$ is the derived category of its heart; but showing it would anyway amount to doing the above computation I guess, by Lurie's recognition principle (Higher Algebra, 1.3.3.7).

If this can help, in my particular situation of interest, $I$ is the category of $\mathbb{Z}$-constructible sheaves on a smooth projective curve $X$ over a finite field and I am looking for instance at $F=\mathrm{Ext}_X^1(-,\mathbb{G}_m)^\dagger$ and $G=\mathrm{Ext}_X^2(-,\mathbb{G}_m)^D/H^1_{ét}(X,-)$ where $(-)^\dagger=\mathrm{Hom}(-,\mathbb{Q})$ and $(-)^D=\mathrm{Hom}(-,\mathbb{Q}/\mathbb{Z})$.

Answer 1

$\newcommand{\Z}{\mathbb Z} \newcommand{\Ch}{\mathrm{Ch}} \newcommand{\Fun}{\mathrm{Fun}}$ Let $\Ch(\Z)$ be the projective model category of chain complexes. It is well known that it presents $D_\infty(\Z)$.

Moreover, $\Fun(I,\Ch(\Z))$ with its projective model structure presents $\Fun(I,D_\infty(\Z))$, and of course $\Fun(I,D^b_\infty(\Z))$ is a full stable subcategory of $\Fun(I,D_\infty(\Z))$, hence its mapping spaces (in fact mapping spectra as well) are the same as the ones in the latter.

But now $\Fun(I,\Ch(\Z)) \cong \Ch(\Fun(I,\Z-\mathrm{Mod}))$, with the projective model structure everywhere, so the mapping spaces can be computed as usual : with projective resolutions (in $\Fun(I,\Z-\mathrm{Mod})$).

So your question is precisely a question of functor cohomology, which is well studied subject, with various techniques.

(note : this is for small $I$. I'm not sure how to make sense of the question when $I$ isn't small)