One of the most famous examples of higher Artin stacks is the stack of perfect complexes.

I recall here the basic stuff:

The idea is that 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 provide an atlas for it, so it turns that this is stack is the quotient of the 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

• chain complexes are objects
• chain maps are 1-morphisms
• chain homotopies are 2-morphisms
• chain homotopies between chain homotopies are 3-morphisms

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...