# Geometric Realization of a Simplicial Category

Let $S:\varDelta^{op}\to (cat)$ be a functor where the category on the right is the category whose objects are categories with cofibrations and morphisms are exact functors(from Waldhausen's paper, Algebraic K-Theory of spaces). Waldhausen talks of the geometric realization of such a simplicial category. In the case of a simplicial set, I know how to construct the geometric realization. However, in this case, $S_n$ is a category. It is not clear to me if he is assuming that this is a small category for each n, in which case we could proceed to construct the geometric realization as in the case of a simplicial set. If each $S_n$ is not a small category, then is it still possible to define the geometric realization of $S$?

• Is this a question about how to interpret something that you are reading? (If so, what?) Or a question about how to make a useful new definition for some purpose of your own? (If so, what purpose?) Jan 5, 2011 at 0:58
• Waldhausen first takes the nerve in order to obtain a bisimplicial set and then takes the geometric realization of that. He doesn't say so very explicitly, but it is clear from his arguments. Compare also with Thomason-Trobaugh, section 1.5. Jan 5, 2011 at 8:00
• Yes, the revision was very good in that it was total! The question now makes sense, and I have changed my downvote to an upvote. Jan 7, 2011 at 11:23