Sheaf of chain complexs glued by chain homotopy equivalences Let $(X,\mathcal O_X)$ be a locally ringed space with an open covering $\mathscr U$. Suppose:

*

*For any $U\in\mathscr U$, we have a chain complex $(C_U, d_U)$ such that $C_U$ is an $\mathcal O_X(U)$-module.

*For any $U,V\in \mathscr U$, we have a chain homotopy equivalence $f_{UV}: C_U|_{U\cap V} \to C_V|_{U\cap V}$

*For any $U,V,W\in \mathscr U$, we have that $f_{VW}\circ f_{UV}$ is chain homotopic to $f_{UW}$.

If I was not mistaken, we can take the homologies and glue them: the homologies $H_*(C_U,d_U)$ for $U\in\mathscr U$ form a $\mathcal O_X$-module on $X$.
However, taking the homologies at the start might lose some information. I was wondering if we can reverse the order. Namely, can we somehow 'glue' $(C_U,d_U)$ in the chain level first and take the homology later? Hopefully this was something well-known.
 A: One way of packing the necessary data is the data of twisting cochain of
Toledo and Tong, see  Duality and Intersection Theory in Complex Manifolds. I..
What do they do: If you have an actual sheaf $\mathcal F$, you can reconstruct
it from restrictions $\mathcal F \vert_{U}$ and isomorphisms
$f_{UV}$ by taking the Čech resolution $C(\mathcal F, \mathscr U)$ (built out of sheaves like $i_{U*}\mathcal F\vert_U$;
see, for example, the stacks)
and then taking its zeroth cohomology, or just leaving it be if you're only interested in the sheaf up to a quasiisomorphism.
Note that for building this Čech resolution one doesn't need the sheaf $\mathcal F$ itself, just restrictions and glueing functions are enough.
The cocycle condition $f_{UW} = f_{UV}f_{VW}$ is needed to ensure that
the Čech differential will square to zero. So the cocycle condition is kind of Maurer-Cartan equation for an appropriately defined dg-algebra
$C(\mathcal{End}(\mathcal F), \mathscr U)$.
A twisting cochain is a Maurer-Cartan element in an appropriately defined
dg-algebra $C(\mathcal{End}(\mathcal C^*), \mathscr U)$. This element
allows one to construct a Čech complex, which will be the totalization of
your $C^*_U$'s.
I don't know of a reference for exactly this fact, and I haven't checked
it thoroughly. From the looks of it, one wants the cover to be locally finite, and one does not necessarily want complexes to be bounded. There is
a lot of literature about the twisting cochains: see, for example, this paper on homotopy limits of dg-categories by Block, Holstein and Wei, as well as other papers of each of these authors.
