Timeline for equivalence in simplicial category
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 22, 2014 at 16:06 | answer | added | user62675 | timeline score: 1 | |
| Dec 21, 2014 at 15:12 | review | Close votes | |||
| Jan 10, 2015 at 3:11 | |||||
| Dec 21, 2014 at 14:57 | comment | added | David White | This question appears to be off-topic. It's about an elementary fact of simplicial categories and has been answered in the comments | |
| Dec 18, 2014 at 0:17 | comment | added | Ilias A. | If you put your second definition i.e. "more generally any hammock of length $n$ for any $n\geq 0$ in $L^{H}\mathcal{C}(X,Y)$ such that any map in the row is an element in W" is the "right" definition since it is equivalent to the proposition "$X$ and $Y$ are equivalent if and only if there exists an isomorphism from $X$ to $Y$ in the associated homotopy category $\pi_{0}L^{H}\mathcal{C}= \mathcal{C}[W^{-1}]$. | |
| Dec 17, 2014 at 23:54 | comment | added | user2664 | Is your proposal the same as saying: two objects $X,Y\in L^H\mathcal{C}$ are equivalent if there exists an element of $L^H\mathcal{C}(X,Y)_0$ and an element of $L^H\mathcal{C}(Y,X)_0$ such that the composition of these 0-simplices (in both directions) are related to the identities by a 1-simplex ? | |
| Dec 17, 2014 at 23:45 | comment | added | user2664 | thanks Fernando, where I can find the definition of $\pi_0S$ for $S$ a simplicial category ? In the nlab I only found the def of $\pi_0X$ for $X$ a Kan complex. | |
| Dec 17, 2014 at 22:48 | comment | added | Fernando Muro | In simplicial categories, morphism objects are simplicial sets, so they do have vertices. For a simplicial category, $\pi_0S$ is not a discrete groupoid, it's a category. | |
| Dec 17, 2014 at 22:39 | comment | added | user2664 | @FernandoMuro: I don't understand what you mean by a "vertex in some morphism simplicial set". Moreover, since $\pi_0S$ is a discrete groupoid I guess you mean "becomes an identity in $\pi_0S$"? | |
| Dec 17, 2014 at 21:30 | comment | added | Fernando Muro | An equivalence in a simplicial category $S$ is a map (i.e.~a vertex in some morphism simplicial set) which becomes an isomorphism in $\pi_0S$. | |
| Dec 17, 2014 at 20:54 | history | edited | David White | CC BY-SA 3.0 |
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| Dec 17, 2014 at 18:54 | history | asked | user2664 | CC BY-SA 3.0 |