Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
If $X$ is discrete of cardinality $\mathfrak{c}$, then it is obviously compactly generated. What I'm saying is that if you carry out the construction as I described in the category of all topological spaces, then $\mathbb{R} X$ will not be a TVS, but if you do it in the category of compactly generated spaces it will be a TVS.
Only after writing my response I noticed that you assumed that all maps are monomorphisms in your condition 2. I don't think I have ever seen this condition in the definition of a saturated class. Clearly, this will not follow from condtion 1. However, I believe that the question you have in mind is independent of whether cofibrations in your model categories happen to be monomorphisms.
@Dmitri That's a good observations, thanks for pointing that out. The two arguments don't seem that different though. The verification of the criterion you mention is essentially equivalent to the verification of the relevant properties of $\mathrm{Ex}^\infty$. That's the most laborious part of my argument, the rest is straightforward. In fact, you can even avoid using $\mathrm{Ex}^\infty$ altogether and use the fibrant replacement arising from the standard small object argument instead. It also preserves filtered colimits and is easier to establish than $\mathrm{Ex}^\infty$.
@Dmitri Thanks for the references. They are both very recent and are about much stronger results There should be some earlier references too, but I can't find any. The argument I gave above is also much more elementary. (And I find it quite amusing that we can prove homotopy invariance of filtered colimits by using fibrant replacements.)
@Dmitri You are absolutely right, in non-proper categories you need levelwise cofibrancy, but since the question was specifically about simplicial sets, I just skipped that remark.
At the moment I can't think of a significantly different argument. What would you count as an argument avoiding homotopy colimits? For example, do you consider Quillen's proof of Theorem A independent of homotopy colimits? It uses diagonals of bisimplicial sets which are homotopy colimits and Bousfield--Kan construction reduces general homotopy colimits to this special case. All these things are intimately related to each other so it may not be possible to separate them completely.
