Model categories and cellular maps

Question

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.

Answer 1

The answer to your question is yes in the case of cellular maps of topological spaces. (I think a similar argument works in the simplicial case, but I doubt the result is true in the CW case.)

There is a model structure on the arrow category in which a square (i.e., a morphism) $$ X_0 \quad \to \quad X_1 $$ $$ \downarrow \quad \qquad \qquad \downarrow $$ $$ Y_0 \quad \to \quad Y_1 $$ (regarded as a map from top to bottom) is a cofibration if and only if $X_0 \to Y_0$ is a cofibration of $\text{Top}$ and the induced map $$X_1 \cup_{X_0} Y_0 \to Y_1$$ is a cofibration of $\text{Top}$. A morphism is a weak equivalence (fibration) iff each of the vertical maps $X_i \to Y_i$ is a weak equivalence (fibration) of $\text{Top}$. (I forget what this model structure is called.)

Now an object $Y_0 \to Y_1$ being cofibrant means that the map from the initial object $\emptyset \to \emptyset$ to it is a cofibration.This is the same thing as saying that $Y_0$ is a cofibrant and $Y_0 \to Y_1$ is a cofibration in $\text{Top}$ with respect to whatever model structure we agree to use on the latter.

If we agree to use the Serre model structure on $\text{Top}$, then $Y_0 \to Y_1$ is a cofibration iff $Y_1$ is obtained from $Y_0$ by cell attachments (or is given by taking a retract of such a thing).

In summary, the cofibrant objects of the arrow category are those (inclusion) maps $Y_0 \to Y_1$ of $\text{Top}$ such that $(Y_1,Y_0)$ is a cellular pair (or is a retract of such a thing).