Where I can find a proof of the following statement: ,,We have two categories: $C$ and $C'$. If they are equivalent, then geometric realizations of its nerves are homotopy equivalent"?
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Geometric realization is a functor, so a functor $C \to C^\prime$ induces a morphism of realizations. Since a natural transformation is a functor $C \times (\bullet \to \bullet) \to C^\prime$, any natural transformation induces a homotopy. So you only need an adjunction for an equivalence of realizations. More generally, there are Quillen's theorems A and B which give an comparison of realizations based on the properties of a single functor.
answered Jan 12, 2018 at 18:26
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$\begingroup$ We have two categories: $C$ and $C'$. You didn't mention about taking a nerve of the category. This is also a functor, which induces a morphism, I'm not wrong? I would like to write a formula for this homotopy. We took a nerve of interval $[0,1]$, which is a morphism between two objects, I'm not wrong? $\endgroup$ Commented Jan 12, 2018 at 18:57
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