# The lisse-etale site and derived algebraic geometry

If one reads say Olsson's book on algebraic stacks or Laumon-Moret-Bailly. The lisse-etale topology is used to define quasi-coherent sheaves and the cotangent complex (or rather cutoff's of the cotangent complex). Now it could well be that I did not read close enough, but my impression is that in Toen-Vezzosi's Homotopical Algebraic Geometry II and Gaitsgory-Rozenblyum's books the lisse-etale topology is not mentioned and both construct contangent complexes and pullbacks. So my question is:

Do you need to consider the lisse-etale topology in the setting of derived algebraic geometry? If not, why?

• One can always define the category of quasi-coherent sheaves on a (derived) stack $X$ as the limit $\varprojlim_{S\to X} Qcoh(S)$ over all affine schemes $S$ over $X$. If $X$ is an Artin stack (resp. $n$-Artin derived stack) then this can be described using the lisse-étale site instead, i.e. take the limit only over smooth morphisms $S \to X$; see e.g. Prop. 1.4.2 here. This is equivalent to the description of Qcoh given in Olsson's book essentially for formal reasons. – AAK Dec 11 '17 at 15:51
• Quasi-coherent sheaves and the cotangent complex can even be described in terms of suitable simplicial hypercovers by derived schemes, analogously to Olsson's results. See Sections 5 and 7 of arXiv:0905.4044, or arXiv:1105.4853 for a summary. (The hard part was showing that suitable hypercovers exist.) – Jon Pridham Dec 12 '17 at 10:28