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 $). 



Homotopy excision and homotopy pushout
1) I assume your three model structures have the same weak equivalences, correct me if I'm wrong. Let $\mathcal C$ be a model category and $I$ a small category, e.g. $I=\bullet\leftarrow \bullet\rightarrow\bullet$ if you're interested in push-outs. The homotopy colimit functor $\operatorname{hocolim}\_I\colon\operatorname{Ho}(\mathcal C^I)\rightarrow \operatorname{Ho}(\mathcal C)$ is simply the left adjoint to the constant functor $\operatorname{Ho}(\mathcal C)\rightarrow\operatorname{Ho}(\mathcal C^I)$. Homotopy categories only depend on weak equivalences, hence homotopy colimits too.


2 & 3) Yes, it is a beautiful result in:


MR1452856 Chachólski, Wojciech A generalization of the triad theorem of Blakers-Massey. Topology 36 (1997), no. 6, 1381–1400.


