I ask this question here since I asked it [here](https://math.stackexchange.com/questions/3310262/the-model-category-structure-on-tmon-and-homotopy-colimits) 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](https://doi.org/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 **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?
> 
> 
> 



The model category structure on $\mathbf{TMon}$
See this question:


[Model Structure/Homotopy Pushouts in topological monoids?](https://mathoverflow.net/questions/11059/model-structure-homotopy-pushouts-in-topological-monoids)


especially the answer by John Francis. 


First of all, monoids have classifying spaces, and so a pushout 
diagram $M\_1\gets M\_0 \to M\_2$ of monoids and monoid maps gives rise to 
a pushout diagram $BM\_1 \gets BM\_0 \to BM\_2$, and the claim is that, 
essentially, 
$$
\mathrm{hocolim}\_{\mathbf{TMon}}( M\_1\gets M\_0 \to M\_2 ) = \Omega (
\mathrm{colim}\_{\mathbf{T}} (BM\_1 \gets BM\_0 \to BM\_2 )) .
$$


