Note: I asked the question below last week on MathSE but received no answer. 


Background:


I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This refers to the fact that they can be glued. 


For instance, suppose that $X$ is a complex analytic space and $P\_1^{\bullet}, P\_2^{\bullet}$ are perverse sheaves defined on the open sets $U\_1, U\_2$ respectively. Then if we are given an isomorphism $\alpha\_{12}: P\_1^{\bullet}|\_{U\_1 \cap U\_2} = P\_2^{\bullet}|\_{U\_1 \cap U\_2},$ then there exists a unique (up to canonical isomorphism) perverse sheaf $P^{\bullet}$ defined on $U\_1 \cup U\_2$ with the property that there are isomorphisms $\beta\_i:P^{\bullet}|\_{U\_i} \to P\_i^{\bullet}$ with ${\beta\_2}|\_{U\_1 \cap U\_2}={(\alpha\_{12} \circ \beta\_1) }|\_{U\_1 \cap U\_2}.$


More generally, if one has an open cover $\{U\_i\}$ of $X$ and perverse sheaves $P\_i^{\bullet}$ of with isomorphisms on the overlaps $U\_i \cap U\_j$ satisfying the co-cycle condition, then this data glue in the usual way. 


My question: 


What goes wrong if one tries to glue ordinary complexes of sheaves? Are there counterexamples showing that the gluing property cannot hold in $C^{\bullet}(Sh(X))$ (the category of sheaves on $X$) or in $D^b(X)$ (the bounded derived category of sheaves on $X$)?



Counterexamples to gluing complexes of sheaves
One can ask whether the derived category forms a stack (of triangulated categories). The answer is no:


Let $S^2$ denote the two-sphere (or $\mathbb{P}^1$ if you prefer) and let $k$ denote a ring of coefficients (e.g. $\mathbb{Z}$). Let $k\_{S^2}$ denote the constant sheaf on $S^2$. Consider a map $\alpha: k\_{S^2}[-2] \to k\_{S^2}$ given by a nontrivial element of $Hom(k\_{S^2}[-2],k\_{S^2}) = H^2(S^2)$. For any proper subset $U \subset S^2$ the restrction map $H^2(S^2) \to H^2(U)$ is zero. Thus if the derived category of complexes of sheaves were a stack then $\alpha$ would be zero, but it isn't! (Of course $S^2$ isn't special ... you just need non-trivial cohomology.)


Now let me get to your question. Suppose that we have $F$ on $U$, $G$ on $V$ and an isomorphism $F\_{U \cap V} = H = G\_{U \cap V}$. Then if C denotes the cocone of the complex


$C \to F \oplus G \to H \stackrel{[1]}{\to}$


(I have omitted lots of $j\_\*$'s) then $C$ will have the property you want, but it won't be defined up to canonical isomorphism. I guess you'll also get intro trouble if you have more complicated covers, because of non-canonicity of cones.


Also, note that you won't be able to get all objects in this way. For example, I think it is impossible to construct the cone of $\alpha$ (where $\alpha$ is as in the first paragraph), by glueing of things defined on proper open subsets.


