In my first comment I meant "$deg(\tau_{i})\leq2$ " and "On the other side, are all the derived nilpotents (in any derived scheme/stack) of homotopical nature?"

Also, thinking about my last sentence it seems to me that broadly speaking the new spaces that derived geometry gives rise to are self intersections spaces (i.e. formal loop and higher dimensional formal loop spaces) and "derived thickenings" by nilpotent extensions (yielding formal disks/higher dimensional formal disks). Does this make any sense?

Thank you for the answer. It is quite concrete an clarifying. The intro of the paper is useful too. However, I am unsure about higher homotopical groups (i.e. $deg(\tau_{i})<-2$). In which situations you can have higher (or abritary high?) homotopical perturbations (you dont need higher dimensional singularities for that, right?)? On the other side if are all the derived nilpotents (of any derived scheme/stack) of homotopical nature? If so, the relevant geometry/topology would be homotopy classes (i.e. maps from the topological space of our scheme/stack to some n-spheres, )