# Constructible derived category and fundamental category

Introduction (may be skipped)

Given a nice topological space $X$, the category of local systems (say over a field $k$) on it is equivalent to the category of representations of its fundamental groupoid. Similar if $X$ is equipped with a stratification the category of constructible sheaves on $X$ is equivalent to representations of its fundamental category (which is defined like the fundamental groupoid but paths are not allowed to go from bigger into smaller strata).

For example if $X=\mathbb P^1$ this shows that the category of constructible sheaves on $\mathbb P^1=\mathbb A^1 \cup pt$ is just representations of the quiver with two vertices and one arrow connecting them.

If one wants to replace constructible sheaves by their derived version, things must be more complicated. The reason is that this category does not only depend on information about $\pi_1$ but also all higher $\pi_i$.

For example in the $\mathbb P^1$ case the constructible derived category is the derived category of representations of a quiver with two points, one arrow in each direction, such that one of their composition vanishes. While I know several proofs of this result, I never really understood it at an intuitive level!

Question

Now first of all I have to admit, that my understanding of infinity categories at a technical level is zero. However, if I get it correctly, in his algebra book Lurie shows that the constructible ($\infty$-)derived category is still equivalent to functors from a higher version of the fundamental category into the ($\infty$-)derived category of vectorspaces. (Really his result is about sheaves of spaces, but I guess there are $k$-linear versions.)

Now my question is:

Can one use Luries results, to get concrete descriptions of the constructible derived categories of simple spaces?

By concrete description I mean writing down a "quiver" (eg a dg-category), such that the constructible dg-derived category is given by representations of this "quiver" in dg-vectorspaces.

For example I would love to see how to do it for $\mathbb P^1=\mathbb A^1 \cup pt$.

Bonus points are awarded if one uses this approach to describe in addtion the category of perverse sheaves.

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Note, regarding perverse sheaves, that you can get such a description using Beilinson's nearby cycles gluing ("How to glue perverse sheaves"). He gives almost exactly this computation at the end of that paper, but for a punctured analytic disk rather than a punctured projective line. The answer is: representations of the quiver with two points and two arrows $u$ and $v$, such that $1 - uv$ and $1 - vu$ are invertible. Obviously that's not the approach you requested, though. –  Ryan Reich May 27 '13 at 16:41
Yes, this is actually one of the proofs I had in mind, when I wrote I never intuitively understood the description. I find constructible sheaves = reps of the fundamental category very convincing. On the other hand Beilinson's glueing is quite intransparent to me. –  Jan Weidner May 28 '13 at 8:36