# What is the application of higher toposes in motivic homotopy theory?

The paper by Raptis and Strunk describes a model for motivic homotopy theory as a model topos. I wonder what this result can lead to or what is the possible development of the relation between motivic homotopy theory and infinity topos? Can anyone suggest an overview or some references please?

• I cannot quite understand this question. Motivic homotopy theory uses essentially the notion of sheaf of homotopy-ish things, so it is naturally connected (at least at a foundational level) with the theory of ∞-topoi (i.e. of sheaves of homotopical objects). The paper you refer to studies a particular ∞-topos, which is connected to the unstable motivic ∞-category (albeit in a slightly opaque way: this is the discussion at the bottom of page 31) – Denis Nardin Jul 3 at 14:24
• @DenisNardin I don't know much of the applications of higher topos. I wonder what is the importance to have a model of infinity topos. – Nicky Jul 3 at 15:08
• The short answer is that working with model topoi rather than ∞-topoi is a matter of personal choice, although ∞-topoi have some advantages in certain settings (e.g. when you start considering families of them). The theory of sheaves in a homotopical setting was fairly fundamental even in setting up motivic homotopy theory, even if they did not use the language of ∞-topoi at the time. What is your background? – Denis Nardin Jul 3 at 16:15
• As pointed out in that paper, it's well-known that the usual $\infty$-category of motivic spaces is a locally cartesian closed presentable $\infty$-category, but not an $\infty$-topos. The contribution of Raptis and Strunk is to provide an alternative $\infty$-category which is an $\infty$-topos and plausibly encodes similar information. As far as I know (though I am not an expert in this area) the promise of this construction has not been fully explored. Note that there are other $\infty$-toposes, such as the etale site, which are related to motivic homotopy theory less directly. – Tim Campion Jul 3 at 16:30
• @DenisNardin I don't mean the choice between the two. The motivic homotopy category constructed by Morel and Voevodsky is not a model topos but this paper shows a way to construct one. I wonder what are some consequences of this model topos? – Nicky Jul 3 at 16:33