Loop space of a category - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:05:36Z http://mathoverflow.net/feeds/question/114553 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114553/loop-space-of-a-category Loop space of a category Mike Shulman 2012-11-26T16:41:59Z 2012-12-06T07:16:23Z <p>This seems like it should be a "standard" thing, and I think I remember even seeing it somewhere, but I can't remember where.</p> <p>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.</p> http://mathoverflow.net/questions/114553/loop-space-of-a-category/114566#114566 Answer by Karol Szumiło for Loop space of a category Karol Szumiło 2012-11-26T18:30:17Z 2012-11-26T23:35:45Z <p>My French is not good enough to be sure about it, but it seems that this paper has the definition you are after.</p> <p>Evrard, Marcel <em>Fibrations de petites catégories.</em> Bull. Soc. Math. France 103 (1975), no. 3, 241–265. (<a href="http://www.numdam.org/item?id=BSMF_1975__103__241_0" rel="nofollow">Numdam</a>)</p> http://mathoverflow.net/questions/114553/loop-space-of-a-category/114694#114694 Answer by Konrad Waldorf for Loop space of a category Konrad Waldorf 2012-11-27T19:34:27Z 2012-11-27T19:34:27Z <p>There is a definition of a <em>loop groupoid</em> of a topological groupoid in </p> <p><a href="http://arxiv.org/pdf/math.AT/0110207.pdf" rel="nofollow">Lupercio, Uribe, "LOOP GROUPOIDS AND TWISTED SECTORS"</a></p> <p>In Section 4 the concept is reduced to the finite group case, where Proposition 4.2.4 presumably shows the property you were asking for.</p> http://mathoverflow.net/questions/114553/loop-space-of-a-category/115502#115502 Answer by Fedotov for Loop space of a category Fedotov 2012-12-05T14:11:55Z 2012-12-06T07:16:23Z <p>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. </p> <p>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 :</p> <p>$\underline{Objects}$ 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 $d\in D$. </p> <p>$\underline{Morphisms}$ 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 $d\in D$.</p> <p>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.</p> <p>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 $Cat_{\Delta}$ </p> <p>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 $S^{1}$.<br> 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 $map(|\mathfrak{C}[k(S^{1})]|, C[C^{-1}])\sim Map(S^{1},N_{\bullet}C[C^{-1}])\sim Map(S^{1}, N_{\bullet} C)=\Lambda N_{\bullet}C$ </p> <p>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 $1\rightarrow C$. </p>