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. Math56(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**.

Question:Are there constructions of the homotopy colimit in

TMonin 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?

freemonoids. $\endgroup$ – Charles Rezk Aug 12 '19 at 16:30