My understanding is that $(\infty,1)$-categories generalise both categories and spaces (topological spaces, Kan complexes or $\infty$-groupoids). One of the most used models for $(\infty,1)$-categories is quasi-categories (aka weak Kan complexes). The fully faithful nerve functor $\mathbf{N}:\mathbf{Cat}\to \mathbf{sSet}$ sends small categories to quasi-categories. So, obviously quasi-categories generalise small catgeoires, but what about large categoires like $\mathbf{Set}$ and $\mathbf{Top}$? The texts I am reading on $(\infty,1)$-categories seem to intentionally ignore some set-theoretic issues as "In these notes we ignore essentially all set-theoretic issues (with the exception of the discussion of locally presentable categories where some care is needed)." in A short course on $\infty$-categories.
Is there a developed model for $(\infty,1)$-categories that gerenalise large categories, in addition to small ones? What is a reference that provide a treatment for such models?
Usually when doing category theory, we work in the settings of Grothendieck universes (or some equivalent set theory).
Then $\mathbf{Set}$ is an abuse for the category of $\mathbb U$-sets for some universe $\mathbb U$ which is clear from the context, it should be denoted $\mathbb U{\text-}\mathbf{Set}$. The axiom of universes then assures that $\mathbb U\text-\mathbf{Set}$ is a $\mathbb V$-small category for a big enough universe $\mathbb V$. Now $\mathbb U\text-\mathbf{Set}$ can be seen, through its nerve, as an object in ${\mathbb V\text-\mathbf{Set}}^{\Delta^{\rm op}}$. The same goes for $\mathbf{Top}$, which is actually a shortcut for the category $\mathbb U\text-\mathbf{Top}$ of topological spaces with $\mathbb U$-small underlying sets.
If you really want to work in a foundationnal setting that only allows two levels (small and large) as NBG, you can still defined what a simplicial class is, and you can still require this simplicial class to fill some horns, leading eventually to a non small quasi-category. You will just not be able to see them as simplicial objects of some category, that's all. You can also adopt another model of $(\infty,1)$-categories as Kan complexes enriched categories.