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 KTheory 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 $S$?

1$\begingroup$ 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?) $\endgroup$– Tom GoodwillieJan 5, 2011 at 0:58

5$\begingroup$ 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 ThomasonTrobaugh, section 1.5. $\endgroup$– Theo BuehlerJan 5, 2011 at 8:00

$\begingroup$ 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. $\endgroup$– Harry GindiJan 7, 2011 at 11:23
1 Answer
As Buehler states in the comments, Waldhausen is taking the nerve degreewise, and then taking the diagonal of the resulting bisimplicial set. This is a model for the homotopy colimit of the simplicial diagram of nerves.
Waldhausen mentions the question of smallness himself in a remark on p. 14 of Algebraic Ktheory of spaces. As he observes, it is only necessary to assume that his categories with cofibrations and weak equivalences are small up to weak equivalence.