Note that category you've described is not locally presentable, but this is not a big deal -- if you use Delta-generated spaces (or some variant thereof) instead of general topological spaces, you're back in the locally presentable world.

On getting by without Vopenka in general:

In Tholen and Rosicky's paper they say (paragraph before Theorem 2.2) that Jeff Smith has claimed that left-determined model structures exist for locally presentable categories without Vopenka's principle. I'm not sure Smith has published his argument.

Existence of a left-determined model structure is essentially an "absolute" version of existence of Bousfield localizations. So some of the existing work on existence of Bousfield localizations under weaker hypotheses may be relevant.

More to the point:

But surely this misses the point, which is that you have a very specific category in front of you and it seems unlikely that the existence of this particular left-determined model structure really depends on set theory. In order to validate this argument, one needs a more detailed understanding of how the model structure works.

Unfortunately Olschok's theory doesn't apply since not every object is cofibrant, if I understand correctly.

It seems remarkable to me that in the related case you discuss, there is a reasonable description of a model structure which can be shown to be left-determined. I would think the way to proceed is to try to imitate whatever happens in that case.

Use an adjunction:

Is there a "geometric realization / nerve" adjunction between the old model category to the new one? I might suspect that if you projectively-induce the old left-determined model structure along such an adjunction, the result may be also left-determined, and in this case you would have a nice description of the model structure.

A Reservation:

I actually don't know whether the Quillen model structure on $\mathsf{sSet}$ is left-determined. If it isn't, then my sense is it's unlikely that the left-determined model structure on multipointed $d$-spaces is actually what you want! I would suspect that the model structure you want is the projectively-induced model structure along some geometric realization adjunction with a more combinatorial category. Then if it turns out that the result is also left-determined, that's great, but if not, it's not a big deal.

Note that category you've described is not locally presentable, but this is not a big deal -- if you use Delta-generated spaces (or some variant thereof) instead of general topological spaces, you're back in the locally presentable world.

On getting by without Vopenka in general:

In Tholen and Rosicky's paper they say (paragraph before Theorem 2.2) that Jeff Smith has claimed that left-determined model structures exist for locally presentable categories without Vopenka's principle. I'm not sure Smith has published his argument.

Existence of a left-determined model structure is essentially an "absolute" version of existence of Bousfield localizations. So some of the existing work on existence of Bousfield localizations under weaker hypotheses may be relevant.

More to the point:

But surely this misses the point, which is that you have a very specific category in front of you and it seems unlikely that the existence of this particular left-determined model structure really depends on set theory. In order to validate this argument, one needs a more detailed understanding of how the model structure works.

Unfortunately Olschok's theory doesn't apply since not every object is cofibrant, if I understand correctly.

It seems remarkable to me that in the related case you discuss, there is a reasonable description of a model structure which can be shown to be left-determined. I would think the way to proceed is to try to imitate whatever happens in that case.

Use an adjunction:

Is there a "geometric realization / nerve" adjunction between the old model category to the new one? I might suspect that if you projectively-induce the old left-determined model structure along such an adjunction, the result may be also left-determined, and in this case you would have a nice description of the model structure.

A Reservation:

I actually don't know whether the Quillen model structure on $\mathsf{Top}$ is left-determined (the one on $\mathsf{sSet}$, of course, is not). If it isn't, then my sense is it's unlikely that the left-determined model structure on multipointed $d$-spaces is actually what you want! I would suspect that the model structure you want is the projectively-induced model structure along some geometric realization adjunction with a more combinatorial category. Then if it turns out that the result is also left-determined, that's great, but if not, it's not a big deal.

Note that category you've described is not locally presentable, but this is not a big deal -- if you use Delta-generated spaces (or some variant thereof) instead of general topological spaces, you're back in the locally presentable world.

On getting by without Vopenka in general:

In Tholen and Rosicky's paper they say (paragraph before Theorem 2.2) that Jeff Smith has claimed that left-determined model structures exist for locally presentable categories without Vopenka's principle. I'm not sure Smith has published his argument.

Existence of a left-determined model structure is essentially an "absolute" version of existence of Bousfield localizations. So some of the existing work on existence of Bousfield localizations under weaker hypotheses may be relevant.

More to the point:

But surely this misses the point, which is that you have a very specific category in front of you and it seems unlikely that the existence of this particular left-determined model structure really depends on set theory. In order to validate this argument, one needs a more detailed understanding of how the model structure works.

Unfortunately Olschok's theory doesn't apply since not every object is cofibrant, if I understand correctly.

It seems remarkable to me that in the related case you discuss, there is a reasonable description of a model structure which can be shown to be left-determined. I would think the way to proceed is to try to imitate whatever happens in that case.

Note that category you've described is not locally presentable, but this is not a big deal -- if you use Delta-generated spaces (or some variant thereof) instead of general topological spaces, you're back in the locally presentable world.

On getting by without Vopenka in general:

In Tholen and Rosicky's paper they say (paragraph before Theorem 2.2) that Jeff Smith has claimed that left-determined model structures exist for locally presentable categories without Vopenka's principle. I'm not sure Smith has published his argument.

Existence of a left-determined model structure is essentially an "absolute" version of existence of Bousfield localizations. So some of the existing work on existence of Bousfield localizations under weaker hypotheses may be relevant.

More to the point:

But surely this misses the point, which is that you have a very specific category in front of you and it seems unlikely that the existence of this particular left-determined model structure really depends on set theory. In order to validate this argument, one needs a more detailed understanding of how the model structure works.

Unfortunately Olschok's theory doesn't apply since not every object is cofibrant, if I understand correctly.

It seems remarkable to me that in the related case you discuss, there is a reasonable description of a model structure which can be shown to be left-determined. I would think the way to proceed is to try to imitate whatever happens in that case.

Use an adjunction:

Is there a "geometric realization / nerve" adjunction between the old model category to the new one? I might suspect that if you projectively-induce the old left-determined model structure along such an adjunction, the result may be also left-determined, and in this case you would have a nice description of the model structure.

A Reservation:

I actually don't know whether the Quillen model structure on $\mathsf{sSet}$ is left-determined. If it isn't, then my sense is it's unlikely that the left-determined model structure on multipointed $d$-spaces is actually what you want! I would suspect that the model structure you want is the projectively-induced model structure along some geometric realization adjunction with a more combinatorial category. Then if it turns out that the result is also left-determined, that's great, but if not, it's not a big deal.