Let C be the category of topological monoids, that is, the category of [monoids](http://en.wikipedia.org/wiki/Monoid_%2528category_theory%2529) in (Top, $\times$). 


1. Can the model category structure on Top (Serre fibrations, cofibrations, weak homotopy equivalence) be transferred to C along the free and forgetful pair of functors ?
2. What are the functorial factorizations in C? Is there a cylinder object in C?


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



Model Structure/Homotopy Pushouts in topological monoids?
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.)


