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Let $(\mathcal{C},W)$ be a category with weak equivalences. One can build from $(\mathcal{C},W)$ its hammock localization $L^{H}(\mathcal{C},W)$ which is a simplicial category $\textit{ie}$ a category enriched in simplicial sets. I wonder if there is a notion of equivalence in $L^{H}(\mathcal{C},W)$ between two objects ?

I can guess a possible answer:

An equivalence in $L^{H}(\mathcal{C},W)$ between X and Y is just an equivalence in $\mathcal{C}$, $\textit{ie}$ a map in $W$, between X and Y, namely a hammock of length 1 in $L^{H}\mathcal{C}(X,Y)_{0}$.

Or more generally any hammock of length n for any $n\ge 0$ in $L^{H}\mathcal{C}(X,Y)_{0}$ such that any map in the row is an element in $W$ (not only the maps that go to the left).

Or more generally any hammock of length n for any $n\ge 0$ which is a k-simplex, $\textit{ie}$ an element of $L^{H}\mathcal{C}(X,Y)_{k}$, such that not only the vertical maps are in $W$ but also all the horizontal maps.

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    $\begingroup$ An equivalence in a simplicial category $S$ is a map (i.e.~a vertex in some morphism simplicial set) which becomes an isomorphism in $\pi_0S$. $\endgroup$
    – Fernando Muro
    Commented Dec 17, 2014 at 21:30
  • $\begingroup$ @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 $\pi_0S$"? $\endgroup$
    – user2664
    Commented Dec 17, 2014 at 22:39
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    $\begingroup$ 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. $\endgroup$
    – Fernando Muro
    Commented Dec 17, 2014 at 22:48
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    $\begingroup$ 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 $\pi_{0}L^{H}\mathcal{C}= \mathcal{C}[W^{-1}]$. $\endgroup$
    – Ilias A.
    Commented Dec 18, 2014 at 0:17
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    $\begingroup$ This question appears to be off-topic. It's about an elementary fact of simplicial categories and has been answered in the comments $\endgroup$
    – David White
    Commented Dec 21, 2014 at 14:57

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Let's think of any simplicial category, $\mathbf{C}$. It's homotopy category is written as $\mathrm{h}\mathbf{C}$, $\mathrm{Ho}(\mathbf{C})$, and even as $\pi_0(\mathbf{C})$ (the last is Fernando's notation). By definition, a map $X\to Y$ in $\mathbf{C}$ that is an isomorphism in $\pi_0\mathbf{C}$ is an equivalence. So in your case, $\mathbf{C}$ is the hammock localization $\mathrm{L}^H(\mathcal{C,W})$. The homotopy category $\pi_0\mathrm{L}^H(\mathcal{C,W})$ is $\mathcal{C}[\mathcal{W}^{-1}]$, the localization with respect to the weak equivalences (obtained by inverting all the morphisms in $\mathcal{W}$); thus, by the above discussion, a map $X\to Y$ in $\mathrm{L}^H(\mathcal{C,W})$ is an equivalence if and only if it is an isomorphism in $\mathcal{C}[\mathcal{W}^{-1}]$. This is, as Fedotov says, equivalent to your second definition.

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