Let $X$ be a locally finite type algebraic stack $X$ (but feel free to pretend it's a scheme) with a presentation as the filtered colimit of finite type open substacks $U\_i$. By descent, at the level of stable derived categories of $\ell$-adic sheaves we have $$D(X) = \varprojlim D(U\_i).$$ Suppose I have two objects $F, G \in D(X)$ and compatible isomorphisms $\phi\_i \colon F|\_{U\_i} \cong G|\_{U\_i}$ for each $i$. My hope is to glue these $\phi\_i$ into a global isomorphism $\phi \colon F \cong G$. I know gluing in derived categories is notoriously tricky; using stable $(\infty-)$ categories one does have the above limit presentation but for general open covers one would in principle need to also specify coherence data; in practice this is so complicated that I have never seen it done along these lines.


However, the open cover in my situation is particularly simple; we could even assume that $U\_1 \subset U\_2 \subset \ldots$. It doesn't seem totally unreasonable that the $\phi\_i$ are enough to glue up a global isomorphism $\phi$ in this case. Is there a way to do this rigorously, or is it a fundamentally erroneous hope?


I'm willing to make some small additional assumptions if necessary to salvage the hope; for example, in my situation of interest $F$ and $G$ will be bounded constructible.



Gluing isomorphism in derived categories along filtered colimit
The hope is reasonable if the open cover really is indexed by $\mathbb N$. Indeed, letting $Spine(\mathbb N)$ denote the simplicial set which can be depicted as $0\to 1\to 2\to ...$ (where the only nondegenerate simplices are the ones I've drawn, in particular there is no $1$-simplex $0\to 2$ !), the inclusion $Spine(\mathbb N)\to \mathbb N$ is a categorical equivalence.


This means two things: 1- To specify a functor out of $\mathbb N$, it suffices to specify objects $X\_0,X\_1,...,X\_n,...$ and morphisms $X\_0\to X\_1, X\_1\to X\_2, ...$; 2- To specify a morphism between two such things, it suffices to specify maps $X\_i\to Y\_i$ as well as commuting squares for adjacent integers, i.e. $i$ and $i+1$. The higher coherence is "automatic" as long as you don't want to impose it.


In particular, for your $\phi$, you really only need compatible isomorphisms $\phi\_i$, and by "compatible", I really only mean that you should provide homotopies $(\phi\_{i+1})\_{\mid U\_i}\simeq \phi\_i$. In fact, if you don't want to know much about $\phi$, it suffices to provide $\phi\_i$'s *with the knowledge that* $(\phi\_{i+1})\_{\mid U\_i}\simeq \phi\_i$. The difference between "with homotopies" and "with the knowledge that there are homotopies" will simply be that in the latter case, you won't specify a single $\phi$; but a class of $\phi$'s which you know is nonempty, whereas if you do specify the homotopies, you will have a well-defined $\phi$.


This is of course very specific to $\mathbb N$, and would fail for most indexing posets $I$ where you indeed have higher coherences that, as you said, often make this sort of venture impossible.


