# Does the Monoid Axiom hold for k-spaces?

In “Algebras and Modules in Monoidal Model Categories” Schwede and Shipley introduced the monoid axiom. If a cofibrantly generated monoidal model category $M$ satisfies this axiom and some smallness hypotheses, then the category $Mon(M)$ of monoids inherits a model structure with fibrations and weak equivalences taken from $M$. In that paper and related papers by the same authors, the monoid axiom is shown to hold on sSet, simplicial functors, simplicial abelian groups, $\Gamma$-spaces, symmetric spectra, $S$-modules, orthogonal spectra, Ch(R), and StMod(R).

The monoid axiom was explored further in Hovey's preprint “Monoidal Model Categories.” Here he showed it to hold on compactly generated spaces and he remarks on page 5 that he doesn't know if it holds on $K$-spaces. The proof he gives for compactly generated spaces fails for $K$-spaces because compact spaces are not ﬁnite relative to closed inclusions, but only relative to closed $T_1$ inclusions. This was about 10 years ago, so I have to ask:

Does the monoid axiom hold on $K$-spaces?

I've read in several papers that there is no known model structure on the category of topological monoids. For instance, here is one written by Vogt in 2012 which makes this claim. I don't know if that's because the monoid axiom is known to fail, or if it just isn't known, or if the obstruction to building a model structure has more to do with smallness than with the monoid axiom. This is what I'm trying to get at.

Is there some known obstruction to $Mon(K$-spaces$)$ being a model category? If not, could experts weigh in on why this is such a hard problem, or on whether or not they think it's true?

My interest in this is for writing the background section of my thesis. I do a lot in my thesis with monoids and commutative monoids, and deal a lot with the monoid axiom. Most of what I do probably won't have an application to categories of spaces, and that was not the motivating application. Still, it would be nice to figure out what my results say in that context (if anything) and that's also what motivated my recent questions on $W$-spaces (see here and here). Incidentally, it was also for this background section that I asked my other question on $K$-spaces and learned that they are not named for Kelley.

Incidentally, I know that some will say this is the wrong question. Instead of strict monoid structures we should care about $A_\infty$ structures. I'm aware of that argument and of the theory in the $A_\infty$ setting, but I have found there are still interesting things to say about strict monoids and strict commutative monoids. So let's restrict attention to that case and only bring in $A_\infty$ as it helps solve the question about strict monoids (e.g. by rectification). Note that this MO question strikes at the difference between $A_\infty$ and strict monoids for spaces.

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Re-reading the Vogt paper which I cited below the first question, I realized that his claim is not as strong as I said. He only claims there is no known Quillen model structure on topological monoids where weak equivalences are (based) maps in $Top^*$ which are not-necessarily-based homotopy equivalences – David White Sep 17 '12 at 20:44
Clark Barwick's answer to a different question (mathoverflow.net/questions/11059) shows that you do have a model structure on topological monoids, but it's for a different category of topological spaces (namely, I think the result he cites needs Top to be Compactly Generated Hausdorff spaces). So you could view my question as trying to make this work for a larger class of topological spaces. If you go with the whole category of topological spaces the question doesn't make sense because you don't have a closed symmetric monoidal category. That's why CG comes in: for function spaces. – David White Sep 17 '12 at 20:50
I suppose I should have been more explicit in my question about which model structure to place on $k$-spaces. If you use the Quillen model structure (weak homotopy equivalences and Serre fibrations) then you're cofibrantly generated but not all objects are cofibrant. If you use the Strom model structure (homotopy equivalences and Hurewicz fibrations) then you are not cofibrantly generated. I'm interested in either case, but more so in the former. There's also the mixed model structure and I'd be interested to know about the monoid axiom there too. – David White Sep 17 '12 at 22:56

It seems to me that there is a relatively simple answer to this question but perhaps I am overlooking something.

The category of K-spaces does satisfy the monoid axiom. (If I read the question correctly K-spaces are what is usually called "compactly generated spaces" and compactly generated spaces are what is usually called "weak Hausdorff compactly generated spaces.") We need to check that a (transfinite) sequential colimit of pushouts of maps of the form $X \times i$ where $X$ is an arbitrary K-space and $i$ is an acyclic cofibration is a weak equivalence. First, observe that $X \times i$ is a weak equivalence and while it is not necessarily a Serre cofibration it is a Hurewicz cofibration.

Now, the conclusion will follow if we know the two following facts.

1. Pushouts of Hurewicz cofibrations which are weak equivalences are again weak equivalences (and of course Hurewicz cofibrations.)
2. (Transfinite) sequential colimits of Hurewicz cofibrations which are weak equivalences are again weak equivalences.

The first one is proven by Boardman and Vogt in Proposition 4.8 (b) in the appendix of Homotopy Invariant Algebraic Structures on Topological Spaces. (The argument doesn't use any fancy point-set topology, in particular separation axioms play no role, so this holds in K-spaces.)

For the second one let's consider a sequence of Hurewicz cofibrations between spaces $(X_\beta \mid \beta < \alpha)$. We want to show that if they are all weak equivalences then so is $X_0 \to \mathrm{colim}_{\beta < \alpha} X_\beta$. First observe that the canonical map from the telescope $\mathrm{Tel}_{\beta < \alpha} X_\beta \to \mathrm{colim}_{\beta < \alpha} X_\beta$ is a homotopy equivalence so it suffices to show that $X_0 \to \mathrm{Tel}_{\beta < \alpha} X_\beta$ is a weak equivalence. By fattening the stages of the telescope slightly we can write it as a colimit of open subspaces homotopy equivalent to the original ones. The conclusion follows since compact spaces are small with respect to open embeddings.

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I'm using the terminology from Hovey's book and preprint, so K-space means compactly open subsets are open, and compactly generated means a K-space which is also weak Hausdorff. Lemma 1.5 from the preprint, shows the category T of compactly generated spaces (in Hovey's language) has the monoid axiom. He proves this using the fact that the generating trivial cofibrations $I^n$×0→$I^n$×I are closed inclusions of strong def retracts. His proof breaks down in K (ie K-spaces) because compact spaces are not finite (categorically). Does your proof secretly use compact $\Rightarrow$ finite anywhere? – David White Feb 15 '13 at 20:46
I can't seem to locate a version of this preprint which contains the lemma you are talking about so I can't comment on its proof. (In the version I've found 1.5 is the definition of smallness.) When I say compact I mean "compact Hausdorff" in the most classical topological sense. Homming out of compact Hausdorff spaces commutes with sequential colimits of open embeddings. In the last paragraph I reduce my colimit to a colimit of open embeddings and in the last line I apply this to spheres in order to see that $X_0 \to \mathrm{Tel}_{\beta < \alpha} X_\beta$ is a $\pi_*$-isomorphism. – Karol Szumiło Feb 16 '13 at 13:29
This of course applies to the Quillen model structure. For the Strøm model structure there is nothing to check since every object is cofibrant so crossing with any space preserves all acyclic cofibrations. – Karol Szumiło Feb 16 '13 at 13:31
It seems to me that for the mixed model structure it is also rather simple. The fibrations of the mixed model structure are the same as those of the Strøm model structure. Thus the acyclic cofibrations of the mixed model structure are the same as those of the Strøm model structure i.e. acyclic Hurewicz cofibrations. Therefore the monoid axiom holds for the mixed model structure since it holds for the Strøm model structure. – Karol Szumiło Feb 19 '13 at 6:47
But again this doesn't seem useful since the mixed model structure is (as far as I understand) not known to be cofibrantly generated. In fact Hovey's remark on page 7 that you mention seems to say something similar about $\mathcal{K}$, namely that we don't have enough smallness properties to construct a model structure on the category of monoids. – Karol Szumiło Feb 19 '13 at 6:53