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Answer by Chris Schommer-Pries for Homotopy limits of quasi-categories

I will address your second question: "one has to prove that the classification diagram functor is sent under this Quillen equivalence to something weakly equivalent to the coherent nerve".The answer is...

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Homotopy limits of quasi-categories

Quasi-categories (or $\infty$-categories, as they are often called) are a very convenient setting for doing abstract homotopy theory. One of their amazing features is the following: Given a diagram of...

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