ct.category theory - Homotopy Pushouts via Model Structure in Top

(I'll assume that in a general model category $mathcal{C}$, $mathrm{Cyl}(X)$ really means: a factorization of $Ato X$ into a cofibration $Ato mathrm{Cyl}(X)$ followed by a trivial fibration $mathrm{Cyl}(X)to X$.)

A sufficient condition on objects for the map in question 1 to be a weak equivalence, is that the objects $X,Y,A$ be cofibrant. (The fact you want to use is the statement due to Reedy (which you can find at the start of the chapter on Proper Model Categories in Hirschhorn's book), that a pushout of a weak equivalence between cofibrant objects along a cofibration is a weak equivalence.)

A sufficient condition on $mathcal{C}$ for the map in question 1 to be a weak equivalence, is that the model category be left proper.

It's an interesting fact that in Top, the this also works if $mathrm{Cyl}(X)$ denotes the "classical" mapping cylinder construction ($X$ union a cylinder on $A$), which isn't necessarily a cofibration in the Quillen model structure. The slickest proof is to use the "excisive triad theorem", as proved by May in A Consise Course in Algebraic Topology, p.79. This also leads to a proof that Top is left proper.