The simplex category $\Delta$ is defined as the category of finite ordinals and order preserving maps. Furthermore, a simplicial set $X$ is defined as a contravariant functor $X: \Delta \rightarrow \text{Set}$.

I am interested in the possibility of generalizing the notion of a simplicial set by considering the category of infinite ordinals $\mathcal{O}$, if such a thing exists, and then defining an $\infty$-simplicial set as $X^{\infty}: \mathcal{O} \rightarrow \text{Set*}$, where $\text{Set*}$ is the appropriate adjustment of the category of small sets $\text{Set}$ such that mapping from objects in the category of infinite ordinals will still satisfy contravariance.

Assuming such a generalization exists, how does this affect the geometric definition of an $(\infty,1)$-category? Is there no longer a capturing of the geometric model desired from simplicial sets or can something more general than an $(\infty,1)$-category be defined? It would also be interesting to see how this would change $(\infty,1)$-functors between simplicial to sets to some other type of functor between $X^{\infty}$ sets. What would natural transformations look like, if they could still be defined properly?