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Since the homotopy category depends only on the 2-skeleton of the simplicial set, the easiest thing to do is to take the 2-skeleton of an ∞-groupoid. For example, let $E$ be the nerve of the contracti …

It depends on what you mean by "all results". Of course results regarding manifolds or vector bundles do not admit statements completely internal to the world of simplicial sets (although most of them …

Well, I guess I can write as an answer what I wrote as a comment. Any pullback square where $C$ is not discrete will yield a counterexample. For simplicity let $B=G=\ast$ and $C=S^1$. Then $E=H=\Omeg …

Let $\mathrm{Ét}_\mathbb{C}$ be the étale $\infty$-topos of schemes over $\mathbb{C}$, that is the $\infty$-categories of étale sheaves of $\infty$-groupoids over $\mathbb{C}$. This contains necessari …

(2) is true (and so (1) is false). To see it, note that every horn $\Lambda^n_i\to S$ to a constant simplicial set must be constant, and so it can be filled by the constant horn $\Delta^n\to S$. Equi …

In the more general version with compact (finitely dominated) fibers, this is called the Becker-Gottlieb transfer. You can find a long list of references on the nlab. Here are a few of them: Becker, …

Yes, you can compute the mapping spaces in ∞-categories by taking the biggest Kan subcomplex of the internal hom. The trick is not to use the Joyal model structure, but instead the model structure on …

It is easier to define the map you're looking for if you adjoint it over and define a map $Map(\Delta[1],X)\to Map(\Delta[1],Map(\Delta[1],X)) = Map (\Delta[1]\times \Delta[1],X)$ Then this comes fr …