Let $\mathcal{C}$ be a simplicially enriched category whose Hom-objects are all Kan complexes. Denote by $N\mathcal{C}$ the homotopy-coherent nerve of $\mathcal{C}$, which is a quasicategory. Suppose I have a simplicial object $X: N\Delta \to N\mathcal{C}$. Can I always find a simplicial object $\tilde{X}: \Delta \to \mathcal{C}$ such that $X$ is weakly equivalent to $N\tilde{X}$?
Simplicial Objects in Quasicategory from Homotopy Coherent Nerve
homotopy-theoryinfinity-categoriessimplicial-categories
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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 under $N^{\Delta}$ of the quotient simplicial functor $\mathcal{C} \to H_2(\mathcal{C})$ yields a map of simplicial sets $h_2(N^\Delta(\mathcal{C})) \to N^D(H_2(\mathcal{C}))$, which one can easily check is an isomorphism of simplicial sets (it suffices to check that it's a bijection on 0-, 1-, and 2-simplices).
The strict (2,1)-categories $|h_2(N^\Delta(\mathcal{C}))|^{\mathcal{D}}$ and $H_2(\mathcal{C})$ are biequivalent; in fact, there is a strict 2-functor $|h_2(N^\Delta(\mathcal{C}))|^{\mathcal{D}} \to H_2(\mathcal{C})$ which is a bijective-on-objects biequivalence. To prove this, note that the composite of the Duskin nerve functor (for strict (2,1)-categories) with its left adjoint sends a strict (2,1)-category $\mathcal{A}$ to its "normal pseudofunctor classifier" $Q\mathcal{A}$, which is a strict (2,1)-category with the universal property that strict 2-functors $Q\mathcal{A} \to \mathcal{B}$ are in natural bijection with normal pseudofunctors $\mathcal{A} \to \mathcal{B}$, for $\mathcal{B}$ a strict (2,1)-category. Moreover, by this universal property, there is a "counit" strict 2-functor $Q\mathcal{A} \to \mathcal{A}$ which one can show is bijective on objects and an equivalence on hom-categories, and hence a biequivalence.
The strict (2,1)-categories $|h_2(C)|^D$ and $H_2(R(|C|^C))$ are isomorphic (if you make a good choice of $R$, e.g. change-of-base along $Ex^\infty$). Indeed, the two functors $|h_2(-)|^D$ and $H_2(|-|^C)$ are naturally isomorphic, and the functor $H_2$ sends the "unit" map $\mathcal{E} \to R(\mathcal{E})$ to an isomorphism (for a good choice of $R$ as above).
I don't know any references for these answers, but these are all straightforward and standard arguments. If you would like me to elaborate on any of these points, I would be happy to.
Both $\def\W{{\bar W}}\W$ and $\def\N{\mathfrak{N}}\N$ are right Quillen functors from the model category of simplicial groups to the model category of reduced simplicial sets (see the original paper by Dwyer–Kan, or Proposition V.6.3 in Goerss–Jardine). Thus, to show that the natural transformation $\W→\N$ is a weak equivalence, it suffices to show that the adjoint natural transformation $\def\L{{\bf L}}\L_\N→\L_\W$ of associated left adjoint functors (denoted by $\L_{(-)}$) is a natural weak equivalence.
The natural transformation $\L_\N→\L_\W$ is a natural transformation of left Quillen functors from simplicial sets to simplicial groups. Since reduced simplicial sets are generated by $\def\Z{{\bf Z}}\W\Z≃S^1$ under homotopy colimits, to show that $\L_\N→\L_\W$ is a natural weak equivalence, it suffices to show that $\L_\N(S^1)→\L_\W(S^1)$ is a weak equivalence of simplicial sets. This is done by a simple direct inspection.
Best Answer
It is easy to construct counterexamples already for the full subcategory $\{[0],[1]\}⊂Δ$. The category $\cal C$ can be constructed by applying the nerve functor to hom-objects of a category $D$ enriched in groupoids. All groupoids have a finite set of objects and at most one morphism between any pair of objects.
The category $\def\id{{\rm id}}D$ has two objects $0$, $1$ and nonidentity morphisms $$s_0:0→1, \quad d_0,d_1:1→0, \quad s_0d_0=s_0d_1=\id_1:1→1, \quad d_0s_0≅d_1s_0≅\id_0:0→0.$$ The latter three 1-morphisms are isomorphic to each other. However, $d_0$ and $d_1$ are not isomorphic to each other.
Now we have an obvious functor $N Δ_{≤1} → N D$ of quasicategories. Observe that the definition of the homotopy coherent nerve functor allows for an isomorphism $d_0s_0≅d_1s_0$. On the other hand, a simplicial object $Δ→D$ does not allow for such an isomorphism. Since any object weakly equivalent to the one constructed above must have $d_0$ and $d_1$ as the images of the corresponding face maps, we see that it is impossible to rectify the homotopy coherent simplicial object to a strict one.