Given a model category $C$, I have two functors $F,G:\mathbb{N}\rightarrow C$, where see $\mathbb{N}$ as sequence category.
Question: Given a natural transformation $J:F\rightarrow G$ and suppose the $J$ induces pointwise weak equivalence then when will the Colimit be weak equivalence?
More specifically on the notes A primer on Homotopy colimits by Daniel dugger in page 10.
https://pages.uoregon.edu/ddugger/hocolim.pdf
I want to know what theorem enables us to conclude from the picture $|X|\rightarrow |Y|$ is weak equivalence.
In general, colimits preserve levelwise weak equivalences when the colimit is also a homotopy colimit. In this case, it is the fact that you are taking a sequential colimit of cofibrations between cofibrant objects in the Kan model structure on simplicial sets that gives you that the colimit preserves levelwise weak equivalences (see e.g. Proposition 17.9.1. in Hirschhorn's Model Categories and Their Localizations for a proof). Alternatively, we can note that such a diagram is Reedy cofibrant, so that the strict colimit also models the homotopy colimit of the diagram because in Hirschhorn's terminology the poset category $\mathbb{N}$ has fibrant constants. (Definition 15.10.1, and see Theorem 15.10.8 for the reason this is important.)