Can you make the category whose objects are pairs of spaces $(X,A)$, and morphisms the obvious diagrams, into a model category? Of course I want this to be done in a meaningful way, that is, agreeing with the adjoint functors $X\to (X,\emptyset)$ and $(X,A)\to X$?

There might be some intuitive reason that it is wrong to expect this, but I don't see it yet.

Thanks!

Can you make the category whose objects are pairs of spaces $(X,A)$, and morphisms the obvious diagrams, into a model category? Of course I want this to be done in a meaningful way, that is, agreeing with the adjoint functors $X\mapsto (X,\emptyset)$ and $(X,A)\mapsto X$?

There might be some intuitive reason that it is wrong to expect this, but I don't see it yet.

Thanks!