All Questions

Q1: Here, on pg $103$, the stack $\text{LocSys}_G(X)$ is defined (for an affine algebraic group $G$, a fixed DG scheme $X$, and a test scheme $S \in \textbf{DGSch}$): $Maps(S, LocSys_G(X)) := \text{...

In the notes by Lurie there seems to be two possible definition for crystals which both makes sense for arbitrary functors $X : \mathrm{CRing}_k \to \mathrm{Set}$. ($k$ here is a field. We probably ...