What is the Isomorphism subspace of the mapping space in an infinity category

Question

When $E$ is a locally free sheaf of rank n on a classical scheme $X$, there is a sheaf $Isom$ on the category $Sch_{X}$ defined as $(S\rightarrow X)\rightarrow Isom_{O_{S}}(O_{S}^{n},E)$. And this establishes a correspondence between $GL_{n}$ tensor on $X$ and locally free sheaf of rank $n$. What I want to do is to extend this correspondence to the derived case when $X$ is a derived scheme.

$\textbf{Question 1:}$ What is the proper replacement of the isomorphism set of two objects in the category of sheaf to the 'space of equivalence' $Isom_{QCoh(S)}(O_{S}^{n},E)$ between two objects in the infinity category of quasi-coherent sheaf on $S$? (Here I take the infinity category $QCoh(S)$ instead of the infinity category $Mod(O_{S})$ because I never see the latter thing being used in the context of derived algebraic geometry) I think this is some subspace of the mapping space $Map_{QCoh(S)}(O_{S}^{n},E)$. But how do we describe this subspace explicitly?

$\textbf{Question 2:}$ Especially, if we have the appropriate definition of the isomorphism space, then for the 'automorphism space' $Aut(O_{S}^{n})\cong Isom(O_{S}^{n},O_{S}^{n})$, is it equivalent to the discrete space $GL_{n}(S)$ (it is discrete because of this post )? Or this equivalence may not be true here?

Answer 1

Question 1: The space of isomorphisms is always a full subspace of the mapping space. In other words, it is a union of connected components of the mapping space.

Question 2: Is $S$ is a classical scheme, then the space of automorphisms of $O^n_S$ in $QCoh(S)$ is discrete, since $O^n_S$ lies in the heart of the t-structure on $QCoh(S)$ (which is the usual abelian category of quasi-coherent sheaves). In general, it is definitely not discrete, but it is still the space of maps from $S$ to the classical scheme $GL_n$. Assuming $S=Spec(A)$, then $$Map_{dSch}(Spec(A),GL_n) = Map_{SCR}(\mathbb Z[x_{ij}][det^{-1}], A) \subset Map_{SCR}(\mathbb Z[x_{ij}], A) = A^{n\times n} = Map_A(A^n,A^n).$$ Here, the inclusion is the full subspace where the determinant is invertible, which is exactly the space of $A$-linear automorphisms of $A^n$. So the point is that polynomial rings and localization have the expected universal property in animated rings.

To answer your motivating question, it is also true that the classical stack $BGL_n$ classifies locally free sheaves of rank $n$ on arbitrary derived schemes. This is the same proof as usual given the fact about $GL_n$ in the answer to Question 2.