I am receiving some very useful comments by private email.
So maybe I should give more details on what I am actually after:

given any category $C$ of fibrant objects, it seems it has a simplicial localization $\hat C$ described roughly as follows:

for $X,A$ two objects of $C$, the 0-cells in the simplicial set $\hat C(X,A)$ are spans

$\array{
&& \hat X &\to& A
\\
& \swarrow
\\
X
}$

with the left leg an acyclic fibration (aka anafunctors, aka generalized morphisms)

and the 1-cells are diagrams

$\array{
&& \hat X_0 &\to& A
\\
&\swarrow& \uparrow && \uparrow
\\
X &\leftarrow&
\hat X_{01}
&\to&
A^I
\\
& \nwarrow & \downarrow && \downarrow
\\
&& \hat X_1 &\to& A
}$

with all the arrows between $X$s acyclic fibrations, and with $A^I$ a path object of $A$ and the maps on the right two boundary evaluation maps out of the path object. So this encodes that the map out of $\hat X_0$ has a homotopty to the map out of $\hat X_1$ after both are pulled back to a joint refinement of their domain.

And analogously for higher simplices.

First question: has this particular kind of construction of simplicial localization in cats of fib objects been considered before? Notice that using the morphisms given by the path object I can make a cosmetic modification to this diagram without changing its content

$\array{
&& \hat X_0 &\to& A
\\
&\swarrow& \uparrow&& \uparrow & \nwarrow
\\
X &\leftarrow&
\hat X_{01}
&\to&
A^I
&\leftarrow&
A
\\
& \nwarrow &\downarrow&& \downarrow & \swarrow
\\
&& \hat X_1 &\to& A
}$

and it begins to look more akin to a hammock. It’s not hammock localization, but supposed to be something more direct using the properties of path objects in cats of fib objects.

Now, if $C$ here is something like locally Kan simplicial presheaves or presheaves with values in some other model of $\infty$-groupoids, then $\hat C(X,A)$ will be a Kan complex, too.

But now assume $C$ is something like (presheaves with values in) weak Kan complexes. Then we are still in a category of fibrant objects, but now we are inclined to allow in the construction above $A^I$ *not* be a path object in the standard sense (not being weakly equivalent to $A$), but be more generally of the form $[D,A]$, where $D$ is some *directed* interval object. The point being that for objects in $C$ behaving (locally at least) like quasi-categories, we don’t want to demand the Hom $SSets$ $\hat C(X,A)$ to have outer fillers.

And there is another thing that makes me wonder, and triggers my question here:

if we drop the condition that $A^I$ in the above hammock-like gadget be weakly equivalent to $A$, then it seems natural to also drop in 1-cells the requirement that all the morphisms between the $\hat X$s be acyclic.

This is kind of remarkable, because the diagram

$\array{
&& \hat X_0 &\to& A
\\
&& \uparrow && \uparrow
\\
&&
\hat X_{01}
&\to&
A^I
\\
& & \downarrow && \downarrow
\\
&& \hat X_1 &\to& A
}$

without any condition on the left vertical morphisms is a bi-brane, namely a span of (generalized) spaces with a gerbe-like thing on each space and a morphism between the pullback of these to the correspondence space.

So I am wondering what’s going on. And if there is some general nonsense on directed path objects which would tell me which conditions to put here on the left vertical morphisms if on the right I allow directed path objects so that the whole thing somehow forms a nice structure.

Oh, I just see a new comment by Mike coming in …

## Re: Question on Homotopical Structure on SimpSet

Well, a different thing to do would be to replace $\Delta^1$ by the nerve $N(J)$ of the walking isomorphism. Then you do get an interval object, in the model-category sense, for both model structures, so that $Kan \hookrightarrow QCat$ should then be an inclusion of categories of fibrant objects.