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Yes, this is true. There are various ways to prove this. Here's the shortest argument I can think of. One direction is easy to prove, so let's prove the other direction. Let $U \colon \mathbf{sSet}^+ …

Yes. This is shown using minimal left fibrations in Cisinski's book Higher categories and homotopical algebra, see the proof of Theorem 5.2.10 therein. This theorem states that the simplicial set $\ma …

Yes, this ought to be true. One way to prove this would be to produce an equivalence to the universal cocartesian fibration defined in Lurie's Kerodon. Here is an outline of such a proof. N.B. I have' …

While I've never encountered a name for the class of simplicial sets you describe (and can't imagine them being worthy of more than a nonce word), I can answer your related question in the affirmative …

Let $A$ be any $(2,1)$-category containing a composable pair of morphisms $f\colon a\to b$ and $g \colon b \to c$, and a $2$-cell $\alpha \colon g \to g$ such that $\alpha f = \mathrm{id}_{gf}$. Then …

There is indeed a simple, explicit description of the left adjoint to the relative nerve functor. This left adjoint sends a simplicial set over the category $C$, i.e. a morphism of simplicial sets $p …

The answer is that every enriched presheaf can be expressed as a weighted colimit of representables. The general definition of a weighted colimit goes as follows. Let $\mathcal{A}$ and $\mathcal{C}$ b …

The simplicial sets $h_2(N^\Delta(\mathcal{C}))$ and $N^D(H_2(\mathcal{C}))$ are isomorphic. To prove this, observe that the universal property of $h_2(N^\Delta(\mathcal{C}))$ applied to the image un …

Such a proof is given in Chapter 3 of Cisinski's book Higher categories and homotopical algebra, see Definition 3.3.7 and Theorem 3.6.1. (Note that Cisinski's proof uses as the interval object not the …