I have three related questions.
I understand homotopy pushouts via the standard model structure on the diagrams - and taking the derived functor of the pushout.
I'm not sure, but I believe that in classical algebraic topology, we use implicitly at least three model structures in Top: Quillen structure, Hurewicz structure and mixed structure.
So, we obtain (at least) three notions of homotopy pushouts, right?
First question: Is there a theorem relating these three types of homotopy pushouts?
Second question: Can homotopy excision be formulated as a result about homotopy pushouts? If so, can we formulate homotopy excision as a result of which (of those three) kind of homotopy pushout?
Third question: Do you know a good reference on this stuff? (Specially, references about homotopy excision using homotopy pushouts).
Remark: "Homotopy excision" is the following result: If $(X,A,B)$ is a excisive triad, such that $(A,C)$ is $(n-1)$-connected and $(B,C)$ is $(m-1)$-connected (with $n\geq 2 $ and $m\geq 1 $), Then $(A,C)\to (X,B) $ is a $(m+n-2)$ equivalence.
It seems clear that all homotopy pushouts, at the HUrewicz model, can be viewed as a excisive triad... And, then, we can formulate the result... I don't know about the other structures. But, even in the Hurewicz model structure, the reformulated version seems to be weaker (since I can't prove that all excisive triad is equivalent to a homotopy pushout - I mean, if $(X, A, B) $ is a excisive triad, I can't prove that X is the homotopy pushout of $A\cap B\to A $ along $A\cap B \to B $).