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

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?

• As an outsider here, I am tempted to suppose that you mean "infinity category" rather than "infinite category" in all occurrences? Might be good to make that change, if appropriate, just to reduce the friction for other readers? Or, if I'm mistaken, I'd be interested to learn what is intended... :) Commented Jul 24 at 23:01
• thank you for your mindful reminding, I have fixed this error.
– Yang
Commented Jul 24 at 23:11

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.