A pushout diagram of derived categories coming from an open cover of schemes

Question

Suppose $X=U\cup V$ is the standard open cover of $X=\mathbb{P}^1$ by two affine lines. The descent theorems say that the diagram (with all arrows restriction maps) $\require{AMScd}$ \begin{CD} D(X) @>>> D(U) \\ @VVV & @VVV \\ D(V) @>>> D(U\cap V)\\ \end{CD} is a pullback (e.g. using the dg category). However, I find this is also a pushout diagram. This is because the restriction $D(X)\to D(U)$ can be regarded as Verdier quotient by the subcategory of $D(X)$ generated by the structure sheaf of the point $X-U$, and similarly for other restriction maps. (Verdier quotient is the cofiber for dg categories)

Is my argument wrong? If it is correct, I wish to know some general account of this phenomenon. Since the category of all dg categories is not stable, I find it weird to have a pushout-pullback diagram.

Answer 1

I don't know if this answers your question, but you seem to have outlined yourself the whole phenomenon yourself.

The point is that while $Cat_{st}$ (or $Cat_{dg}$) is not itself stable, it has a large supply of bicartesian squares, namely those corresponding to localization sequences $C\to D\to D/C$. Therefore, as soon as you have a pullback square where both horizontal legs are Verdier quotients (or maybe Karoubi quotients if you are working up to idempotent-completion), it will also be a pushout square.

As far as I know, we are still lacking a proper context for this phenomenon, but there is hopefully a notion of "$2$-stable $(\infty,2)$-category" which encodes and axiomatizes this and other nice but not-quite-stable behaviours of $Cat_{st}$ and its friends ($Cat_R$ for ring spectra $R$ and so on). Some attempts are being made, but there is no proposed "correct definition" of what this is (as far as I know the places in the literature where this is mentioned all say that their definitions are "temporary and only because they work for the current purposes").