Is there a good nerve-like functor from simplicial objects in categories to simplicial sets which takes level-wise equivalences of categories to weak equivalences? 


To give this some context, I'd like to extract a simplicial set from the Waldhausen S-construction applied to a category with cofibrations, and I realized that my standard way of taking a nerve is for simplicial categories (i.e. simplicial objects in categories for which the objects form a constant simplicial set), and this doesn't clearly apply to the S-construction.



Nerves of simplicial objects in categories/Waldhausen's S-construction
Here is an idea. Try the homotopy coherent nerve. (This was originally introduced, sort of, by Boardman and Vogt in a topological context and was formulated for simplicially enriched categories (and please do not use `simplicial category' as it is ambiguous!) by Cordier in 1980. The H.c. nerve is related to the bisimplicial nerve by using the codiagonal of Artin and Mazur. (which has been mentioned in several of my answers!!!). Some details of the H.C. nerve as discussed in the nLab entry on that and there are links to elsewhere. A chatty discussion can be found in Kamps and Porter, (again that has been mentioned before :-))!


Hope this helps.


[Edit] I should mention the papers


M. Bullejos and A. Cegarra, On the Geometry of 2-Categories and their Classifying Spaces , 
K-Theory, 29, (2003), 211 – 229.


M. Bullejos and A. M. Cegarra, Classifying Spaces for Monoidal Categories Through Geometric Nerves , Canadian Mathematical Bulletin, 47, (2004), 321–331. 


A. Cegarra and J. Remedios, The relationship between the diagonal and the bar constructions 
on a bisimplicial set , Topology and its Applications, 153, (2005), 21 – 51. 


A. Cegarra and J. Remedios, The behaviour of the $\overline{W}$-construction on the homotopy theory of bisimplicial sets , Manuscripta Math., 124, (2007), 427 – 457, ISSN 0025-2611.


some of which may help and that there is related discussion in the Menagerie and in Lurie's HTT.


