Fix a scheme $Z$, and consider a category whose objects are schemes $X$ equipped with a closed immersion $Z\to X$.
Obviously, a morphism $f:X\to Y$ should commute with the respective closed immersions $Z\to X$ and $Z\to Y$. But I would like to impose an additional condition: we should have $f(X\setminus Z) \subset Y\setminus Z$. Equivalently, $f$ should be bijective on $Z$.
Has anyone encountered this category? Is it equivalent to, or a special case of, something well-known? I am particularly interested in understanding the case where all the schemes are affine.
The motivation here is that $X$ and $Z$ may have nice properties—for example, quasi-compactness—that $U=X\setminus Z$ does not. By adding a stratum to $U$, we can use the study of these nicer schemes to draw conclusions about $U$. But a morphism $X\to Y$ of stratifications of $U$ and $V$ respectively should restrict to an honest morphism $U\to V$.
I think this is already interesting in the category of pointed manifolds: A morphism $M\to N$, where $M$ and $N$ are manifolds with a distinguished point $\ast$, should restrict to a morphism $M\setminus\{\ast\} \to N\setminus\{\ast\}$. For example, if we consider the complex plane inside the Riemann sphere, then $z\mapsto z$ is the restriction of such a map, but $z\mapsto e^z$ is not because it does not extend to $\infty$.