I am a beginner in derived algebraic geometry and I am trying to develop some visual and geometrical intuition about derived schemes (and stacks), or more precisely about the new geometrical phenomena that they introduce.



It seems to me that (broadly speaking) the new spaces that derived geometry gives rise to are:


  1. (Possibly higher dimensional) Loop spaces. They arise as self-intersections: e.g. see comments of [J.Pridham](https://mathoverflow.net/questions/350883/what-or-how-are-the-new-spaces-of-derived-algebraic-geometry#comment879218_350883) and the answer of DamienC ([1](https://mathoverflow.net/a/354496)) below.
  2. Derived infinitesimal disks/Formal neighbourhoods. Originated by nilpotent extensions. See for example the definition 1.1 in [Vezzosi - A note on the cotangent complex in derived algebraic geometry][1].
  3. **QUESTION: What else?** (See also an answer of DamienC below ([2](https://mathoverflow.net/a/355642))). I think that while 1 and 2 are already present in derived schemes other phenomena require derived stacks. 

I would like to see more examples that have some geometrical interpretation. There are cases of derived stacks for example in [Toen - Higher and derived stacks: A global overview][2]. Such examples include the derived stack of rank $n$ local systems over some topological space (and the derived moduli stack of vector bundles), derived linear stacks, and the derived stack of perfect complexes. 
However I am unable to obtain a geometrical meaning for this examples.

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**EDIT**: *What is the geometrical interpretation of the higher homology groups in (for example) the derived stack of vector bundles over a projective variety $\mathbb{R} \underline{{Vect}_{n}}(X)$?* 

According to [this paper from Toen-Vezzosi][3] some of the motivation for this derived stack comes from the will to build a smooth moduli space (unlike the underived case). 
When $X=S$, a *smooth projective surface*, they claim that the tangent space at a point $E$ is: 
$T_{E} \mathbb{R} \underline{\operatorname{Vect}}_{n}(S) \simeq-H^{2}(S, \underline{E n d}(E))+H^{1}(S, \underline{E n d}(E))-H^{0}(S, \underline{E n d}(E))$.
However here the $H^{2}$ term (which is the derived part) seems to come from the fact of  $S$ is $2$-dimensional and not from any singularity or self-intersection (which seems strange to me)

If you look at the example of ([2](https://mathoverflow.net/a/355642)), which is quite similar  (I think it is the derived stack of local systems),  the $H^{2}$ term appears when  you take into account the self intersection of $0$ in $\mathbb{A}^{1}$, (i.e. a truly derived structure).

What I am misunderstanding here?


  [1]: http://www.dma.unifi.it/~vezzosi/papers/derivedintctgtcplx.pdf
  [2]: https://hal.archives-ouvertes.fr/hal-00772928/document
  [3]: https://hal.archives-ouvertes.fr/hal-00773012/document