Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its hammock localization in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion:

If there is an initial object in $\mathbf{C}'$, then $\mathbf{C}$ has a contractible classifying space.

Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its hammock localization in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion:

If there is an initial object (up to homotopy) in $\mathbf{C}'$, then $\mathbf{C}$ has a contractible classifying space.

By initial up to homotopy I mean an object $c$ so that $\mathbf{C'}(c,d)$ is contractible for all other objects $d$.

To any small category $\mathbf{C}$ one can associate a category $Z\mathbf{C}$ of zigzags which shares the same objects, but where a morphism from $x$ to $y$ looks like a finite diagram in $\mathbf{C}$ of the form: $$ x \to x_0 \gets x_1 \to \cdots \gets x_k \to y,$$

with two such diagrams identified whenever they differ only by identity morphisms, and composition defined by the obvious concatenation. I'm looking for a reference to (something like) the following theorem:

If there is an initial object in $Z\mathbf{C}$, i.e., an object with a unique class of zigzags to all other objects, then $\mathbf{C}$ has a contractible classifying space.

Maybe the first question to ask might be: is the result even true? I've convinced myself that it is, but in what appears to be an incredibly inelegant fashion. Is there a slick proof in the literature?

Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its hammock localization in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion:

If there is an initial object in $\mathbf{C}'$, then $\mathbf{C}$ has a contractible classifying space.