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Can somebody help me with a reference showing that the homotopy semicosimplicial totalization of a cosimplicial space is homotopy equivalent to its usual homotopy totalization? Is it because the inclusion of the semisimplicial category into the simplicial category is a homotopy left cofinal functor?

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Having thought a little bit I proved that the inclusion of the semisimplicial category to the simplicial category is a homotopy left cofinal functor. But it must be a very standard thing and written somewhere. Can somebody help me with a reference? –  Victor Oct 15 '10 at 19:37
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I agree this should be standard, but I've only seen the proof in one place. See Lemma 6.5.3.7 of Lurie's Higher Topos Theory.

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Hi Hiro! Glad to hear from you and thanks for your reference. But of course this thing is older than Lurie: E. Dror, W. G. Dwyer. A long homology localization tower. Comment. Math. Helv. 52 (1977), no. 2, 185210. –  Victor Sep 23 '11 at 19:13
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