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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.

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What about taking the nerve in each simplicial degree and then taking the diagonal of the resulting bisimplicial set? –  K.J. Moi Jun 20 '11 at 16:21
    
I think that does something slightly different, but it does helpfully rephrase the question: Taking the levelwise nerve puts us in bisimplicial sets, and takes levelwise equivalences to levelwise homotopy equivalences, i.e. to weak equivalences (w.e.) in the Bousfield-Kan or Reedy structures. Taking the diagonal of a bisimplicial set preserves w.e. in the Moerdijk structure (by definition), but not necessarily in the BK or Reedy structures. So, I'm looking for a functor from bisimplicial sets to simplicial sets which preserves BK or Reedy w.e. –  Jesse Wolfson Jun 20 '11 at 16:48
    
If we consider a bisimplicial set as a simplicial object in simplicial sets then a levelwise weak equivalence induces a weak equivalence on diagonals, right? Sorry if I'm missing the point here. –  K.J. Moi Jun 20 '11 at 17:39
    
All simplicial sets cofibrant (in the Quillen model structure). Do you mean with the Joyal model structure? –  David Carchedi Jun 20 '11 at 17:51
    
@KJ, I don't think that a levelwise w.e. necessarily induces a w.e. on diagonals. That was my point about the different model structures on bisimplicial sets. @David, thanks, you're absolutely right. –  Jesse Wolfson Jun 20 '11 at 18:38
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2 Answers

up vote 3 down vote accepted

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.

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I don't know if it will be a full answer to your question but there is a (diagonal) model structure on the category of simplicial objects in $\mathbf{Cat}$ (denoted by $\mathbf{sCat}$) such that a map $F_{\bullet}:C_{\bullet}\rightarrow D_{\bullet}$ between two object in $\mathbf{sCat}$ is a fibration (weak equivalence) iff the diagonal of the nerve (taken level-wise) of the corresponding level-wise groupoids is a fibration (weak equivalence) of simplicial sets, i.e.

$ diag ~\mathrm{N}_{\bullet}\mathbf{iso}~F$ is a fibration (a weak equivalence) of simplicial sets.

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