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Let $X= S^1 \wedge S^1$ be a wedge of circles. Then $X$ admits a natural stratification $\mathcal{S}$ as a union of two disjoint open intervals $I_1, I_2$ and a point $\{*\}$.

Let $D_{\mathcal{S}}(X)$ be the dg category of sheaves of $\mathbb{C}$-vector spaces which are constructible with respect to $\mathcal{S}$. (i.e. a dg enhancement of the classical derived category of sheaves of $\mathbb{C}$-vector spaces constructible with respect to $\mathcal{S}$). Let $\mathcal{F}_1, \mathcal{F}_2$ be (respectively) the constant sheaf with stalk $\mathbb{C}$ in degree $0$ on $I_1 \cup \{*\}$ and on $\{*\} \cup I_2$. Let $\mathcal{A} \subset D_{\mathcal{S}}(X)$ be the full subcategory with objects $\mathcal{F}_1, \mathcal{F}_2$.

A dg category is said to be formal if it is equivalent (as a dg category) to its cohomology category.

Question: is $\mathcal{A}$ formal?

Remark: I am actually hoping that the answer is ``no", but I have no particular reason to believe this.

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  • $\begingroup$ I'm pretty sure that $hom(F_1,F_2) \simeq 0$ and reversely. It follows that this is the direct sum of the full subcategories on $F_1$ and $F_2$, each of which is formal (it's the one-object dg-category on cochains on the circle). I find it easiest to think about all of this in terms of exit-paths: S-constructible sheaves on $X$ are the same as representations of $Exit_S(X) \simeq (I_1 \leftleftarrows p \rightrightarrows I_2)$, a category with three objects and morphisms as indicated. [cont.] $\endgroup$
    – Aaron Mazel-Gee
    Commented Oct 5, 2021 at 17:23
  • $\begingroup$ Note that $F_2$ is a pushforward (equivalently a right Kan extension), so by adjunction you can take the pullback of $F_1$ to get $hom(F_1,F_2) \simeq hom( \mathbb{C} \rightrightarrows 0 , \mathbb{C} \rightrightarrows \mathbb{C} ) \simeq 0$. $\endgroup$
    – Aaron Mazel-Gee
    Commented Oct 5, 2021 at 17:25
  • $\begingroup$ Maybe I've drastically misunderstood something, but if I have two constant sheaves $F_1, F_2$ , each supported on a different circle, then surely $\operatorname{Hom}(F_1, F_2)=\mathbb{C}$ (nothing derived here). The space of morphisms from $F_1$ to $F_2$ is global sections of the sheafy-hom, which in this case is a skyscraper. Did I misunderstand what you are saying? $\endgroup$
    – Laurent Cote
    Commented Oct 6, 2021 at 17:18
  • $\begingroup$ $hom(F_1,F_2)\simeq hom(\mathbb{C}\rightrightarrows 0, \mathbb{C}\rightrightarrows \mathbb{C})$ is correct, but that's not equivalent to $0$ -- there's a hom in degree 1. $\endgroup$
    – Ben G
    Commented Oct 6, 2021 at 20:39
  • $\begingroup$ Ah, thanks @BenG! So the category is not a direct sum, after all. $\endgroup$
    – Aaron Mazel-Gee
    Commented Oct 7, 2021 at 18:11

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