The nerve functor $N:Cat\to sSet$ has a left adjoint, namely the categorisation $C$. In fact there is a natural isomorphism $\epsilon: CN\to Id$ and $N$ is a full embedding.So if I start with a simplicial set $X$, I can generally not hope for $NCX$ to be isomorphic to $X$, but can I say something about the geometric realisation? In other words, is there a nice relation between $BCX$ and $|X|$?
In my example $X$ is the simplicial ring $Gl(R^\bullet)$ where an $n$ simplex is an element of the general linear group over $R[T_0,\cdots,T_n]/(\sum T_i=1)$. $CX$ should then be the category with objects $Gl(R)$ and morphisms $Gl(R[T])$ (source and target given by $T=0,1$, respectively). I tried to make sense of the nerve of this category in comparison to $X$ but failed.
The underlying motivation is a discussion of the hocolim of a functor $F:K\to Cat$ for a small category $K$ and $Cat$ the category of small categories (I am reading Thomasons paper). If I understand it correctly there we are actually taking the hocolim of the nerves over all $F(k)$, $k\in K$ and see then that this hocolim itself is the nerve of a category. Now this category comes with a universal mapping property into other categories. Eventually we obtain a universal mapping property out of hocolim into nerves of arbitrary categories. This seems somehow unnatural since the whole construction worked simplicially and not categorial. Yet I was unable to go back to arbitrary simplicial sets.
Edit Let me get this straight: Given is a functor $\mathcal C\to CX$ for a small category $\mathcal C$ and a simplicial set $X$. I want to show that it induces a map of simplicial sets $N\mathcal C\to X$. If I take the adjoint functors $CSd^2$ and $Ex^2N$ instead ($Sd$ is subdivision and $Ex$ its adjoint) then I get $$Hom(\mathcal C,CSd^2X)\cong Hom(Ex^2N\mathcal C,X)$$ I have an natural acyclic cofibration $N\mathcal C\to Ex^2N\mathcal C$ and a natural map $CSd^2X\to CX$. So all I need to show is that my initial map $\mathcal C\to CX$ factors over $CSd^2X$. Is this correct, is there a good way to show the last bit?
