I ask this question here since I asked it here on Math.SE, and got no answers after a week of a bounty offer.
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 the paper
R. Schwänzl and R.M. Vogt, The categories of $A_\infty$- and $E_\infty$-monoids and ring spaces as closed simplicial and topological model categories, Arch. Math 56 (1991) pp 405–411, doi:10.1007/BF01198229.
Cofibrations are the morphisms which have the appropriate lifting property.
I wish to understand what the homotopy colimit of a diagram of topological monoids is. One way of approaching this is to take the colimit of the cofibrant replacement of the diagram in question. This involves (firstly) understanding cofibrations in TMon.
I struggle with this. I really have no intuition for what a cofibration in this category is at all. This is the first thing preventing me from understanding homotopy colimits of diagrams in TMon.
Are there constructions of the homotopy colimit in TMon in the case of simple diagrams? For example, if a diagram has morphisms which are all inclusions on the level of topological spaces? Or if the diagram is Reedy? Or under any other sufficiently nice assumptions?