Let $(\mathcal{C},W)$ be a category with weak equivalences, one can built out from it 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 Y and X, namely a hammock of lenght 1 in $L^{H}\mathcal{C}(X,Y)_{0}$.

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

Or more generally any hammock of lenght 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.

Best

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.