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The question is very unlikely to have a satisfactory answer in the level of generality asked. Let $X$ and $Y$ be the two spaces with isomorphic homotopy groups, and let $LX$ and $LY$ denote their loca …

For (1)-(2), look at work of Carles Casacuberta. He has lots of good examples. His paper with Chorny on the orthogonal subcategory problem has an example for your (2), on the last page. This paper of …

First, in any model category, weak equivalences are closed under homotopy pullback. That's the whole point of the "homotopy" in "homotopy pullback." In your question, you are missing a step. General …

The idea you're looking for is called "fibrewise localization". It's defined in Dror Farjoun's book "Cellular Spaces, Null Spaces, and Homotopy Localization", and also in Hirschhorn's book (since you …

While preparing the paper Left Bousfield localization without left properness, I learned another example, due to Voevodsky. It's example 3.48 in his paper Simplicial radditive functors. This is an exa …

Yes, they are the same. In order to prove it, we should show that they have the same new weak equivalences (this is enough, because both have the same cofibrations). Have a look at Barwick's paper On …

Yes. This is Theorem 3.3.20 in Hirschhorn's book, if by $T$ you mean the left derived maps of $L(S)$, e.g. cofibrant replacements of the maps $L(S)$.

I have an unpublished note that proves Barwick's claim. Aspects of this story have appeared in some papers of mine with Michael Batanin, including one we published in the proceedings of the 2015 CRM c …