Model Structure/Homotopy Pushouts in topological monoids?

Question

Let $\mathsf C$ be the category of topological monoids, that is, the category of monoids in $(\textsf{Top}, \times)$.

1. Can the model category structure on $\textsf{Top}$ (Serre fibrations, cofibrations, weak homotopy equivalence) be transferred to $\mathsf C$ along the free and forgetful pair of functors ?

2. What are the functorial factorizations in $\mathsf C$? Is there a cylinder object in $\mathsf C$?

I'm mostly interested in computing a homotopy pushout in the category $\mathsf C$, so any ideas how to do that would be helpful too.

Answer 1

The answer to question #1 is yes. You can use Kan's theorem on lifting model structures (11.3.2 in Hirschhorn's book) to obtain a model structure on $C$ such that the weak equivalences (resp., fibrations) are those morphisms of topological monoids that are weak equivalences (resp., Serre fibrations) on the underlying space. The cofibrations here don't seem to admit an elementary description, sadly. (This model category is Quillen equivalent to the category of simplicial monoids. Depending on the particular nature of your homotopy pushout, you may find it easier to compute there.)

Answer 2

It's easy to describe the group completion of a pushout, up to homotopy. If $Y\leftarrow X \rightarrow Z$ is the diagram of topological monoids, then the group completion of the pushout is equivalent to $\Omega(BY\amalg_{BX} BZ)$, the based loop space of the pushout of the diagram of pointed spaces $BY \leftarrow BX \rightarrow BZ$.

If your monoids are grouplike to begin with then their pushout is also grouplike, and this gives you an answer.

Answer 3

Clark Barwick's answer is excellent and you should accept it. This is more of an addendum. The category Top is cofibrantly generated, so $\mathcal{C} =$ Mon(Top) is also cofibrantly generated. The key paper is by Schwede and Shipley, and gives conditions on a model category $\mathcal{M}$ such that Mon$(\mathcal{M})$ is a model category. In the special case of $\mathcal{M}$ cofibrantly generated it explains how to get your hands on the cofibrations of Mon$(\mathcal{M})$. See Theorem 4.1 on page 8. Of course, now that you have your hands on the fibrations, trivial fibrations, cofibrations, and trivial cofibrations question (2) is also answered. A nice reference for relating the cylinder object to the functorial factorizations is Hovey page 9.

Furthermore, every element in Top is fibrant, so the paper above gives you even stronger results, which may help you with your computations. See remark 4.5 on page 10.

The authors also wrote a second paper giving further results. It's here.