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 $S$?
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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
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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 Thomason-Trobaugh, section 1.5. $\endgroup$– Theo BuehlerJan 5, 2011 at 8:00
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$\begingroup$ 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
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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 K-theory 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.
