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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 $d$.

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  • $\begingroup$ Do you have an interesting example where $Z \mathcal{C}$ has an initial object? The only ones I can think of are categories $\mathcal{C}$ with a unique strict initial object such as $\mathbf{Set}$, and those are contractible simply by virtue of having an initial object. $\endgroup$
    – Zhen Lin
    Aug 13 '14 at 13:28
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    $\begingroup$ Do you mean for x→y←x to be the identity on x in ZC when the right- and left-pointing arrows are the same? If so, then your ZC is a groupoid and only has an initial object when it has a single object. If not, then powers of $x \to y \leftarrow x$ are probably all distinct, and I would expect $Z\mathbf{C}$ to not have an initial object unless $\mathbf{C}$ is a single object groupoid. (I haven't thought this through very carefully.) $\endgroup$ Aug 13 '14 at 14:16
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    $\begingroup$ I think Omar is right. You seem to be describing the Gabriel-Zisman localization of $\bf C$ at all arrows, i.e. the groupoidification of $\bf C$. Hence $Z\bf C$ is the fundamental groupoid of the nerve of $\bf C$, and it has an initial object iff it is trivial, but this says nothing about the higher homotopy groups of $\bf C$, unfortunately. $\endgroup$ Aug 13 '14 at 14:33
  • $\begingroup$ @ZhenLin certainly I don't mean to use this for anything like $\mathbf{Set}$. But tons of examples can be generated by taking various copies of $x \to x_0 \gets x_1 \cdots$ and gluing them all at $x$ for instance. The category so obtained does not have an initial object, but the zigzag category contracts to $x$. $\endgroup$ Aug 13 '14 at 15:20
  • $\begingroup$ @ViditNanda I think that up the equivalence relation that you made for id morphisms (which I'm not sure to understand), the category $Z\mathsf{C}$ looks like the underling category of the Dwyer-Kan Hammock localization $L_{\mathsf{C}}\mathsf{C}$ at all morphisms of $\mathsf{C}$. More precisely $Z\mathsf{C}=L_{\mathsf{C}}\mathsf{C}_{0}$ i.e. you take only the 0 cells of $L_{\mathsf{C}}\mathsf{C}$ which is now an ordinary category. $\endgroup$
    – Ilias A.
    Aug 13 '14 at 17:36
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The answer to your new question is positive. Dwyer and Kan prove in Calculating simplicial localizations that the hammock localization ${\bf C}’=L^H\bf C$ is weakly equivalent to the standard simplicial localization $L\bf C$. In Simplicial localizations of categories, they had previously showed that $L\bf C$ and $\bf C$ have weakly equivalent nerves. The nerve functor preserves weak equivalences of simplicial categories with the same objects (see Simplicial localizations of categories 1.4 (vii)), so it is enough to prove that $NL\bf C$ is contractible. Since you’re taking hammock/simplicial localization w.r.t. all arrows in $\bf C$, $L\bf C$ is a simplicial groupoid by its very definition. The existence of a homotopy initial object, which is transferred to $L\bf C$ by the weak equivalence above, implies that all simplicial sets of morphisms in $L\bf C$ are contractible. Therefore, $L\bf C$ has the same nerve as the trivial groupoid on the same set of objects as $\bf C$, which is of course contractible.

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First of all Muro's answer is perfect. This a long comment which I hope it will be helpful, it is more about intuition. I think Rezk's answer Group completion theorem is closely related to your question. For any (small category) $\mathsf{C}$ and any sub category $\mathsf{W}$, Dwyer and Kan defined a simplicial localization $L_{\mathsf{W}}\mathsf{C}$ (i.e. it a simplicially enriched category which same set of objects as $\mathsf{C}$) which has the following properties:

There is an obvious functor $\mathsf{C}\rightarrow L_{\mathsf{W}}\mathsf{C}$ such that

a) $\mathsf{C}[\mathsf{W}^{-1}]\simeq \pi_{0}L_{\mathsf{W}}\mathsf{C}$

b) For any morphism $a\rightarrow b\in \mathsf{W}$, we have $$Map_{L_{\mathsf{W}}\mathsf{C}}(b,c)\rightarrow Map_{L_{\mathsf{W}}\mathsf{C}}(a,c)$$ and $$Map_{L_{\mathsf{W}}\mathsf{C}}(c,a)\rightarrow Map_{L_{\mathsf{W}}\mathsf{C}}(c,b)$$ are equivalences of simplicial sets.

c) The $N_{\bullet}\mathsf{C}$ is equivalent to the (coherent) Nerve $N_{\bullet}L_{\mathsf{W}}\mathsf{C}$

In the case where $\mathsf{W}=\mathsf{C}$, the simplicial category $L_{\mathsf{W}}\mathsf{C}$ is an $\infty$-goupoid.

An other categorical way to construct $L_{\mathsf{W}}\mathsf{C}$ is the following: Take the push out in the model category $\mathbf{Cat}_{\mathbf{sSet}}$ (the category of all small simplicial categories):

$$colim[\overline{1}_{\mathsf{W}}\leftarrow 1_{\mathsf{W}}\rightarrow \mathsf{C}]$$

where $1_{\mathsf{W}}$ is described as follows, it is a disjoint union indexed by morphisms in $\mathsf{W}$ of the category of the form $\bullet\rightarrow\bullet$, and the category $\overline{1}_{\mathsf{W}}$ is described as follows, it is a disjoint union indexed by morphisms in $\mathsf{W}$ of simplicial category $I$, the simplicial category $I$ has two objects and all its mapping spaces $Map_{I}$ are contractible. Moreover, the morphism $\overline{1}_{\mathsf{W}}\leftarrow 1_{\mathsf{W}}$ is a cofibration in $\mathbf{Cat}_{\mathbf{sSet}}$. Then the simlplicial category $L_{\mathsf{W}}\mathsf{C}$ is zigzag equivalent to $colim[\overline{1}_{\mathsf{W}}\leftarrow 1_{\mathsf{W}}\rightarrow \mathsf{C}]$ as simplicial categories. Applying the (coherent Nerve) to the diagram $colim[\overline{1}_{\mathsf{W}}\leftarrow 1_{\mathsf{W}}\rightarrow \mathsf{C}]$ (which is a homotopy colimit actually, since $\mathbf{Cat}_{\mathbf{sSet}}$ is left proper) we obtain a homotopy pushout in the category of simplicial sets (with Quillen model structure). This is not obvious since the nerve is not well behaved with respect to colimites. Then you conclude that the nerve $N_{\bullet}C\rightarrow N_{\bullet}L_{\mathsf{W}}\mathsf{C}$ is a weak equivalence of simplicial sets since $N_{\bullet}1_{W}\rightarrow N_{\bullet}\overline{1}_{W}$ is a trivial cofibration in simplicial sets (which is left proper). Your first question (before you edited it) can be summarized to the effect of the nerve functor to the following diagram $C\rightarrow [L_{C}C]_{0}\rightarrow L_{C}C \rightarrow \pi_{0}L_{C}C= C[C^{-1}]$ where the last morphism is the localization morphism (functor). So, only the map $N_{\bullet}C\rightarrow N_{\bullet}L_{C}C$ is an equivalence.

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