Homotopy excision and homotopy pushout

Question

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 $).

Answer 1

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.

Answer 2

There are old-fashioned classical ways to think about excision, which can easily be translated into model theoretical language of homotopy pushouts as desired. Any excisive triad can be approximated up to weak equivalence of triads by a CW triad, where a CW triad (X;A,B) is just a CW complex X that is the union of subcomplexes A and B (theorem on page 77 of "A concise course in algebraic topology"). A CW triad is not excessive, but it is homotopy equivalent to the obvious double mapping cylinder, which is excessive (lemma on page 78). Note that excision in any homology theory is obvious for a CW triad since A/A cap B is isomorphic as a CW complex to X/B. The Blakers-Massey theorem is given an easy homotopical proof from this CW approximation result on page 85, opus cit. No originality is claimed: I'm pretty sure I learned the idea from Boardman (probably unpublished).

Chapter 17 of "More concise algebraic topology" has a reasonably thorough treatment of the three model structures and their comparison. The mixed model structure is due to Cole. (However `More Concise' unfortunately has a mistake, taken from a different paper of Cole, in the proof of the axioms for the Strom model structure; a nice paper of Barthel and Riehl, "On the construction of functorial factorizations for model categories" explains and corrects the mistake.)