One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff:

> We fix a function $b: \mathbb{Z} \rightarrow
 \mathbb{N}$ which is zero outside of finitely many values, and we let
> $\operatorname{Perf}^{\leq b} $ be the higher stack whose value on a
> scheme $X$ is the $\infty$ -category of perfect complexes $C$ over $X$
> such that at any point $x \in X$, we have $h^{i}\left(C_{x}\right)
  \leq b(i) .$  So $\operatorname{Perf}^{\leq b} $ is an $n$ -stack,
> where $n$ is the length of the interval on which $b$ is nonzero.
> 
> It is proven that the Buchsbaum–Eisenbud schemes (or similarly the varieties of complexes) provide an atlas for
> it, so this is stack turns to be the quotient of these  varieties of complexes by the  equivalence relation identifying two complexes
> which are quasi-isomorphic.

My question is basically: what are geometrically the $n$-morphisms in this stack?

I guess that we have that:
 - chain complexes are
 objects
 - chain maps are 1-morphisms
 - chain homotopies are 2-morphisms
 - chain homotopies between chain homotopies are 3-morphisms (and so on...)

I guess that we need $n$ terms in our chain complex to have non-trivial $n$-morphisms of this kind but I don't know why...

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**EDIT**: It seems according to [this][1], [this][2] and [this paper][3], each point of the stack would be a complex $C$ with automorphism and higher automorphisms groups (in which the $n$-(auto)morphims that I wonder about live in) equal to:

$\pi_{1}\left(\operatorname{Perf}^{\leq b} , C\right) \simeq A u t(C) \quad \pi_{i}\left(\operatorname{Perf}^{\leq b} ,C\right) \simeq E x t^{1-i}(C,C)\: n > i>1$

which reflects the fact that "morphisms between complexes of amplitude in $[a, b]$ has homotopies and higher homotopies up to degree $n-1$, or equivalently that the $\infty$-category of complexes of amplitude in $[a, b]$ has $(n-1)$ -truncated mapping spaces." (cause according to [this][4] $ E x t^{1-i}(C,C)=Hom\left ( H^{i}(C) , H^{1}(C)\right )$ can be nonzero only for $1<i<n$)

My understanding is that these $Hom\left ( H^{i}(C) , H^{1}(C)\right ), i>1$ should correspond to higher mapping spaces **but I don't know the exact form of them and specially I don't know how this reflects in terms of the $n$-atlas (the varieties of complexes)**


  [1]: https://arxiv.org/pdf/1401.1044.pdf
  [2]: https://arxiv.org/pdf/math/0503269.pdf
  [3]: https://arxiv.org/pdf/math/0604504.pdf
  [4]: https://stacks.math.columbia.edu/tag/06XS