TITLE: The localization of the span category
QUESTION [5 upvotes]: Suppose one has a model category $C$ with its class of weak equivalences $W$. It is possible to form a separate homotopical category $(C_{\operatorname{span}},W_{\operatorname{span}})$ which has objects given by the objects of $C$ and morphisms from $x$ to $z$ given by spans $x  \overset{\simeq} \twoheadleftarrow y \twoheadrightarrow z$ where the first arrow is an acyclic fibration and the second arrow is a fibration. The class of weak equivalences $W_{\operatorname{span}}$ is given by the spans consisting of two acyclic fibrations. The composition of two spans is defined by pullback.
In case it is not obvious, the purpose of the fibration requirement is to ensure that the composite of two such spans is still a span with the backwards arrow an equivalence.
It is well known that we may take a Hammock localization of a relative category in order to obtain a simplicially enriched category. The process involves considering commutative diagrams of zig-zags. Is it the case that the Hammock localization of $(C,W)$ is equivalent to the Hammock localization of $(C_{\operatorname{span}},W_{\operatorname{span}})$?

REPLY [2 votes]: Here's half an answer.
See Prop 5.2 of Dwyer and Kan's Function Complexes in Homotopical Algebra for a proof that the hammock localization of $C$ is the same as the hammock localization of the non-full subcategory of fibrations in $C$.
Then the remaining question is whether introducing the backwards weak equivalences affects the hammock localization. I think in order to see this you will have to spell out exactly what bicategory of spans you're using (what its universal property is), but I think the fact that this doesn't affect the hammock localization should follow from considering the universal properties involved.