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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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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?) –  Tom Goodwillie Jan 5 '11 at 0:58
    
I voted to close as well. –  Dr Shello Jan 5 '11 at 6:47
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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. –  Theo Buehler Jan 5 '11 at 8:00
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Dr. Shello, you can't vote to close because you don't have enough reputation. –  Steven Gubkin Jan 5 '11 at 11:22
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+1, good revision. –  S. Carnahan Jan 5 '11 at 21:00

1 Answer 1

up vote 6 down vote accepted

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.

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