Homotopy Pushouts via Model Structure in Top - MathOverflow

Homotopy Pushouts via Model Structure in Top

2010-01-08T20:07:00Z

As far as I know, one way to take a homotopy colimit in a model category is to replace (up to acyclic fibration) all arrows in the diagram with cofibrations, and take the strict colimit of the resulting diagram.

In Top with the model structure given by Serre fibrations, cofibrations, and weak equivalences, if one wants to obtain a homotopy pushout of the diagram $X \leftarrow A \rightarrow Y$, it is "enough" to replace only one of these arrows with a cofibration: that is, there is a natural map (by the universal property of the pushout) $Cyl(X)\cup_A Cyl(Y) \to Cyl(X) \cup_A Y$ that is a homotopy equivalence of spaces.

Question 1: What conditions on the model category $\mathcal{C}$ (or objects $X,Y,A$) will guarantee that the natural map $Cyl(X) \cup_A Cyl(Y) \to Cyl(X) \cup_A Y$ is a weak equivalence?

Question 2: This question is less precise, but if the map above is a weak equivalence, does that mean $Cyl(X) \cup_A Y$ is a good model for the homotopy pushout?

Answer by Reid Barton for Homotopy Pushouts via Model Structure in Top

2010-01-08T20:25:40Z

Question 1: The model category $\mathcal{C}$ should be left proper, i.e. the pushout of a weak equivalence along a cofibration is again a weak equivalence. (Dually, there is a notion of right proper.) Top is left proper, as is any model category in which every object is cofibrant, such as SSet. There is some information on this notion of properness at the nlab, and I think it's also discussed more thoroughly in Hirschhorn's book Model Categories and their Localizations (and probably many other places).

Question 2: Yes. People often say that a square in a model category is a homotopy pushout square if the induced map from the (strict) pushout of a cofibrant replacement (meaning cofibrant objects and maps) of the "initial" three objects to the last object is a weak equivalence, and that is the case here.

Answer by Charles Rezk for Homotopy Pushouts via Model Structure in Top

2010-01-08T20:30:43Z

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