### Short Version (the question)


Let $\text{Cat}$ be the category of (small) categories and $\text{sSet}$ the category of simplicial sets. There is a functor $\Gamma:\text{sSet}\to \text{Cat}$ that takes every simplicial set to its [category of simplices](http://ncatlab.org/nlab/show/category+of+simplices). This functor has a right adjoint, say, $T$.



> 
> **Question 1:** Given a simplicial set $X$, is it true that $T(\Gamma(X))$ is weakly equivalent to $X$?
> 
> 
> **Question 2:** Is the unit $\epsilon : 1\to T\circ \Gamma$ of the adjonction a weak equivalence of simplicial sets?
> 
> 
> 


### Long Version (background, motivation, partial results etc.)


To fix notation, let $\Delta$ be the simplex category whose objects are the non-empty finite ordinals $$[n]=(0<1<...<n)$$ and morphisms are the weakly monotone maps. There is the nerve functor 
$$
N:\text{Cat}\to \text{sSet}
$$
$$
N(C)\_n = Func([n],C)
$$


Where $[n]$ is viewed as a poset category. The nerve functor has a left adjoint $\tau\_1$ and the composition $\tau\_1 \circ N$ is naturally isomorphic to the identity. On the other hand, the composition $N\circ \tau\_1$ loses a lot of information and in particular is not weakly equivalent to the identity. One solution to this "problem" is to apply barycentric subdivision twice before taking $\tau\_1$. Thomason has shown that this does not lose homotopic information and used this to transport the Quillen model structure of $\text{sSet}$ to $\text{Cat}$.


Another "solution" is to use a different way to pass from $\text{sSet}$ to $\text{Cat}$, given by the category of simplices functor $\Gamma:\text{sSet}\to \text{Cat}$. It is known that there is a natural weak equivalence $N(\Gamma(C))\to C$. Hence, $\Gamma$ is in some vague sense inverse to $N$ on the homotopical level. Now, the functor $\Gamma$ itself has a right adjoint which is given by the following explicit formula
$$
T(C)\_n = Func(\Delta /[n], C)
$$


Where we use the notation $\Delta/[n]$ for the over-category and the simplicial maps come from the functoriality of the over-category construction (namely, by pullback). 


Now, one way one might hope to answer the first question affirmatively, is to compare $N(\Gamma(C))$ with $T(\Gamma(C)$. In fact, one might hope that $N(C)$ is weakly equivalent to $T(C)$ for every category $C$. A candidate for such a weak equivalence is given by the Bausfield-Kan map $\Delta/[n] \to \Delta$ which can be viewed as a morphism of cosimplicial categories. This gives a map $N(C)\to T(C)$ which one might hope to be a weak equivalence.


Some evidence that this might be true in general, come from the special case where $C$ is a poset. In this case, every functor $\Delta/[n] \to C$ factors uniquely through the "posetization" of $\Delta/[n]$, which is the poset category of non-degenerate simplices of $\Delta^n$. The nerve of this category is just the barycentric subdivision of the n-simplex $\Delta^n$. Thus, we obtain


$$
T(C)\_n = Func(\Delta/[n],C) = Func(Pos(\Delta/[n],C) = Hom(sd(\Delta^n),N(C))= Hom(\Delta^n, Ex(N(C)) = Ex(N(C))\_n
$$


and it follows that $T(C)$ is just the Kan extension of $N(C)$ which is known to be weakly equivalent to $N(C)$ (this was a bit sketchy, but hopefully clear enough)



Does the right adjoint of the category of simplices functor is "homotopicaly inverse" to the category of simplices functor?
This indeed true and is discussed in depth by Latch, Thomason and Wilson in a paper called *Simplicial Sets from Categories*. Your question is answered in Corollary 4.7 and relies on Theorem 4.1, the main result of the paper which says that $N \to T$ is indeed a weak equivalence. The argument can also be adapted to give a very neat combinatorial proof that $\mathrm{id} \to \mathrm{Ex}$ is a weak equivalence.


