I am trying to understand the homotopy colimit of a diagram of topological monoids, and whether there is an explicit construction of this object (even in the case of simple pushout diagrams).
Let TMon denote the category of well-pointed topological monoids which have the homotopy types of cell complexes.
TMon can be equipped with a model category structure where the fibrations and weak equivalences are the (Serre) fibrations and weak equivalences of the underlying topological spaces - this comes from section $3$ of this paper by R. Schwänzl and R. Vogt. Cofibrations are the morphisms which have the appropriate lifting property.
I wish to see what the homotopy colimit of a diagram of topological monoids is - one way of getting the homotopy colimit is to take the colimit of the cofibrant replacement of a diagram. This involves (firstly) understanding cofibrations in TMon.
Understanding cofibrations isn't easy - a large chunk of the above paper is indeed dedicated to getting a better grip on them. My intuition tells me that they should be the inclusions which also then respect the monoid structure "up to homotopy", but that's really not based in anything at all.
My question is whether there is a known construction of the homotopy colimit of such a diagram?
Alternatively, are there constructions in the case of simple diagrams? For example, if a diagram has morphisms which are all inclusions of topological spaces? Or if the diagram is Reedy?
EDIT:
As remarked by Max in the comments, a cofibration in this category had better indeed respect monoid multiplication on the nose, or else it isn't a map of monoids at all (even less a cofibration). As such, I really have no intuition for what a cofibration in this category is at all.