A question came up on MSE and it generated, for me, the following question: When looking at the maps of CW/cell/simplicial complexes do the cellular/simplicial maps have a model theoretic interpretation? Are they cofibrant objects in some model structure on the arrow category of topological spaces? (feel free to replace topological spaces with anything reasonable such as CW/cell/simplicial complexes)

For clarity, by arrow category I mean that the objects are the morphisms of Top and the morphisms of the arrow category are commutative squares. I am happy to accept that no good model structure lives on this category and happy to have this category suitably replaced.

Maybe it is obvious, and my apologies if it is such.

Thanks for your time.