# The model category structure on $\mathbf{TMon}$

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.

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?

• Cofibrations in $TMon$ are going to be more specific: they will be retracts of "cellular maps", i.e., of maps built by iteratively attaching cells. A cell will be something of the form $F(S^{n-1})\to F(D^n)$, where $F$ is the free monoid on a space. Thus, the "simplest" examples of cofibrant objects are discrete free monoids. – Charles Rezk Aug 12 '19 at 16:30
• Thanks for your comment - I'm a bit of a beginner with this. Is it possible for you to elaborate a little on why cofibrations should be retracts of such maps, or why a "cell" is what you give above? – Matt Aug 13 '19 at 10:52
• It's an example of a general formalism for constructing model category structures by "lifting". Here we are lifting the usual model structure on spaces to $TMon$: so the we's and fib's of TMon are "detected" by the forgetful structure to spaces. Formally, the left adjoint $F$ to the forgetful functor will preserve cofibrations and trivial cofibrations. In particular, it must supply a large collection of cofibrations in TMon, which can be used to "build" all cofibrations. – Charles Rezk Aug 13 '19 at 16:52
• Key phrases here are "cofibrantly generated model category" and "tranferred model structure": ncatlab.org/nlab/show/cofibrantly+generated+model+category, ncatlab.org/nlab/show/transferred+model+structure – Charles Rezk Aug 13 '19 at 16:53

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, $$\mathrm{hocolim}_{\mathbf{TMon}}( M_1\gets M_0 \to M_2 ) = \Omega ( \mathrm{colim}_{\mathbf{T}} (BM_1 \gets BM_0 \to BM_2 )) .$$