Timeline for Turning simplicial complexes into simplicial sets without ordering the vertices
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Dec 9, 2022 at 3:58 | history | edited | LSpice | CC BY-SA 4.0 |
Name of "here", while this is on the front page
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| Dec 8, 2022 at 23:04 | answer | added | Dan Ramras | timeline score: 3 | |
| Apr 18, 2019 at 8:55 | answer | added | Johannes Ebert | timeline score: 5 | |
| S Apr 16, 2019 at 19:03 | history | bounty ended | CommunityBot | ||
| S Apr 16, 2019 at 19:03 | history | notice removed | CommunityBot | ||
| Apr 9, 2019 at 22:38 | comment | added | Omar Antolín-Camarena | @მამუკაჯიბლაძე Yes, very appropriate! After all $X(K)_n$ is precisely the set of morphisms of simplicial complexes from the n-simplex to $K$. | |
| Apr 9, 2019 at 21:38 | comment | added | მამუკა ჯიბლაძე | Concerning terminology and notation - it might be natural to call elements of your $X(K)$ singular simplices of $K$ and, accordingly, denote $X(K)$ by $\operatorname{Sing}(K)$. | |
| Apr 9, 2019 at 20:25 | comment | added | Omar Antolín-Camarena | @მამუკაჯიბლაძე Yes, the complex of injective words on $n$ letters has the homotopy type of a wedge of $D_n$ copies of $S^{n-1}$. (I think the original reference for the homology is Farmer, Cellular homology for posets. Math. Japon. 23 (1978/79), no. 6, 607–613; and for the homotopy type Bjorner, Wachs, On lexicographically shellable posets. Trans. AMS, 277(1):323–341, 1983, but I learned it from Randal-Williams, Homological stability for unordered configuration spaces. Q. J. Math. 2011 Dec 6;64(1):303-26.) I don't about your simplicial set for simplicial complexes of more than 1 simplex | |
| Apr 9, 2019 at 18:56 | comment | added | მამუკა ჯიბლაძე | Yes you are right. It only has four nondegenerate simplices: two vertices, and two edges pointing in the opposite directions, this is clearly a circle. For higher $n$ rank of the top homology grows like $1,2,9,44,265,...$ - seems to be the derangement numbers | |
| Apr 9, 2019 at 18:19 | comment | added | Omar Antolín-Camarena | @მამუკაჯიბლაძე I don't think the simplicial set you describe is homotopy equivalent to the complex you start with. For example if $K$ is a single simplex of dimension $n$, then I believe your simplicial set is the simplicial set obtained by freely adjoining degenerate simplicies to the semi-simplicial set known as the "complex of injective words on $n+1$ letters", which is not contractible. Specifically, for $K$ a single interval your simplicial set has the homotopy type of $S^1$. | |
| Apr 8, 2019 at 17:45 | comment | added | მამუკა ჯიბლაძე | If one only reads the question title, one foolish way that comes to mind is this: allow only neighboring repetitions. That is, if $x_i=x_{i+j}$ then $x_i=x_{i+k}$ for all $0<k<j$ too. I wonder if this is also equivalent to $K$... | |
| S Apr 8, 2019 at 17:15 | history | bounty started | Omar Antolín-Camarena | ||
| S Apr 8, 2019 at 17:15 | history | notice added | Omar Antolín-Camarena | Draw attention | |
| Mar 27, 2019 at 7:46 | comment | added | Andrea Gagna | No problem, glad to be of help! | |
| Mar 27, 2019 at 4:13 | comment | added | Omar Antolín-Camarena | I added your proof to the note I wrote, @AndreaGagna, I hope that's OK. | |
| Mar 27, 2019 at 0:12 | comment | added | Omar Antolín-Camarena | I think you mean $X(K) = v^* v_! X_{\le}(K)$ (you were missing a $v^*$), and I think I agree, @AndreaGagna. | |
| Mar 26, 2019 at 22:39 | comment | added | Andrea Gagna | As for the notation, I think the simplicial set $K(X)$ is strictly linked to symmetric simplicial sets. If $\Upsilon$ denotes the category of symmetric simplices, there is a canonical funtor $v \colon \Delta \to \Upsilon$ which induces a Quillen equivalence pair $(v_!,v^*)$ between the category of presheaves (see §8.3). It seems to me that $K(X)$ is precisely $v_!(K_{\leq}(X))$ which in turn is precisely your $E \otimes_{\Delta} K_{\leq}(X)$. Moreover, you show that the unit $1 \to v^*v_!$ is always a weak homotopy equivalence. | |
| Mar 26, 2019 at 21:32 | history | asked | Omar Antolín-Camarena | CC BY-SA 4.0 |