# Homotopy excision and homotopy pushout

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 $A\cap B \to B$).

• I'm not sure if this is exactly what you're seeking, but it's worth noting that these three model structures are Quillen equivalent (via the identity functor) as can be seen from the nLab article on the Strom model structure. With that in mind, this MO question gives one possible answer for your first question: mathoverflow.net/questions/82813. Does that help at all for the application you have in mind? – David White Apr 10 '13 at 20:23
• For (3) the references at the bottom of the nLab article are great: ncatlab.org/nlab/show/homotopy+limit. There was also a MO question previously asking for references for homotopy colimits: mathoverflow.net/questions/454/references-for-homotopy-colimit. I haven't read those closely enough to know if they include anything about (2), because I've never really thought about (2) before and I'm not sure it's true. Where's the cofibrant replacement in homotopy excision? – David White Apr 10 '13 at 20:26
• Since the inverse question - "when the homotopy pushout squares in the Quillen model are homotopy pushout squares in the Hurewicz model" - seems to have only a trivial answer, I think the question 1 should be ignored. – Fernando Apr 10 '13 at 21:30
• Perhaps, I should be more clear in question 3. When you have a excisive triad, the pushout of $(A\cap B)\to B$ along $(A\cap B)\to A$ is $X$. Thus a excisive triad may be seen as a special kind of pushout. If you have a homotopy pushout diagram, I guess you can replace for a homotopy equivalent excisive triad. So, the hypothesis of the homotopy excision may be replaced by "a homotopy pushout in which the morphisms are (n-1) equivalence and (m-1) equivalence". Then you have something about the homotopy pullbacks... .... – Fernando Apr 10 '13 at 21:31
• @David White: I am afraid the Strom/Hurewicz model structure on topological spaces is not Quillen equivalent to the Quillen model structure. One only gets a Quillen adjunction between these model structures. – Ricardo Andrade Apr 10 '13 at 21:55

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.

• The Hurewicz model structure has homotopy equivalences as weak equivalences, while the other ones have weak homotopy equivalences as weak equivalences. Thank you for the article. I will take a look. – Fernando Apr 10 '13 at 21:16
• I see, I call that model structure the Strom model structure. – Fernando Muro Apr 10 '13 at 21:32

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.)

• Presumably "excessive" should be "excisive". Or perhaps topologists have a better sense of humour than algebraists. – dhagbert Apr 11 '13 at 10:01
• Inadvertent, but I'll leave it as is :) – Peter May Apr 11 '13 at 13:23
• It was a helpful answer too! Thank you – Fernando Apr 16 '13 at 21:58