Loop space of a category

Question

This seems like it should be a "standard" thing, and I think I remember even seeing it somewhere, but I can't remember where.

Let $C$ be a small category. Is there a category $\Lambda C$ whose nerve (or classifying space) is a model for the free loop space of the nerve (or classifying space) of $C$? Probably the objects of $\Lambda C$ should be something like zigzags of morphisms in $C$. A reference would be the best thing to hear.

Answer 1

My French is not good enough to be sure about it, but it seems that this paper has the definition you are after.

Evrard, Marcel Fibrations de petites catégories. Bull. Soc. Math. France 103 (1975), no. 3, 241–265. (Numdam)

Answer 2

I will try to answer the question. As I said in a comment, the Thomason model structure on $Cat$ is not simplicial model structure. Let $C$ be a small category, we will view it as a topological category. Denote by $C[C^{-1}]$ the topological category where we invert all maps of $C$ such that $C\rightarrow C[C^{-1}]$ is a cofibration of topological categories, then the coherent nerve $N_{\bullet}C\rightarrow N_{\bullet}C[C^{-1}]$ induces a weak equivalence of simplicial sets. Notice that $C[C^{-1}]$ is an infinity groupoid.

Let $C$ be a cofibrant topological category. The mapping space $map(C,D)$ in the model category of topological categories is given by the (standard) nerve of the following $HOM(C,D)$ category :

$\underline{Objects}$ are topological functor $F:C^{op}\times D:\rightarrow Top$ such that for any $c\in C,$ $F(c)$ is equivalent to a representable functor $D(d,-)$ for some $d\in D$.

$\underline{Morphisms}$ in this category are natural transformation $H:F\rightarrow G$ such that $F(c,d)\rightarrow G(c,d)$ is a weak equivalence for all $c\in C$ and $d\in D$.

Let $S^{1}$ a simplicial model for a circle. Let $k: sSet\rightarrow sSet$ the cocontinues Joyal functor which take $\Delta^{n}$ to the nerve of the groupoid with $n+1$ objects and only one isomorphism between any two objects.

Recall that $\mathfrak{C}: sSet\rightarrow Cat_{\Delta}$ is the left quillen adjoint to the coherent $N_{\bullet}$ betwen the joyal model structre on $sSet$ and the Bergner model structure on $Cat_{\Delta}$

Now $k(S^{1})$ is a simplicial set, and the cofibrant topological category $|\mathfrak{C}[k(S^{1})]|$ is an infinity groupoid and its cohenrent nerve is equivalent to $S^{1}$.
The finial result is that $HOM(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])$ is a model for $\Lambda C$, since the nerve of $HOM(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])$ is equivalent to $ map(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])\sim Map(S^{1},N_{\bullet}C[C^{-1}])\sim Map(S^{1}, N_{\bullet} C)=\Lambda N_{\bullet}C$

N.B. The only point that I did not explained is the construction of $C[C^{-1}]$. Let 1 be the category with two objets a and b and a unique morphisms from $: f:a\rightarrow b$. Let $\widehat{1}=|\mathfrak{C}[k(\Delta^{1})]|$, then $ C[C^{-1}]$ is the pushout $colim (\sqcup_{mor C} \widehat{1}\leftarrow \sqcup_{mor C} 1\rightarrow C )$ i.e., for each porphism of $C$ there is a map $1\rightarrow C$.