Internal categories in simplicial sets

A category object internal to simplicial sets is the same as a Segal space in which the Segal conditions hold on the nose instead of merely up to weak equivalence. In other words, a category is something whose nerve has unique horn fillers instead of merely contractible spaces of fillers.

The above category objects generate a full sub-(relative category) of Rezk's relative category of complete Segal spaces. As I explain below, Barwick and Kan's work proves that the inclusion of this sub-(relative category) induces an equivalence of homotopy theories.

Barwick and Kan construct a nerve functor $N$ from small relative categories to simplicial spaces. The key point is that anything in the image of this nerve is a category object in the above sense.

Their nerve functor $N$ has a left adjoint $K$, but they also consider a second functor $M$ from simplicial spaces to relative categories. The functors $M$ and $N$ are inverse equivalences of homotopy theories in the sense that there is a zigzag of natural weak equivalences $$NMX \rightarrow NKX \leftarrow X$$ for any simplicial space $X$, and a natural weak equivalence $$MNY \rightarrow Y$$ for any relative category $Y$.

If one restricts the domains of $K$ and $M$ to consist only of category objects, the above natural weak equivalences remain intact. Thus the Barwick+Kan homotopy theory of relative categories is equivalent to the theory of category objects in simplicial spaces.

It appears that Geoffroy Horel has solved this problem completely:

Geoffroy Horel, A model structure on internal categories in simplicial sets, Theory and Applications of Categories 30 No. 20 (2015) pp. 704–750 (journal page, arXiv:1403.6873)