Feb22 comment Descent properties of spaces Sorry, I was probably making some mistake in my previous comment, because I wasn't able to reconstruct that argument later. Feb22 answered Descent properties of spaces Feb19 comment Descent properties of spaces For your second problem, this seems to follow by applying Theorem 7.1(b) twice to the maps $X' \to Y$ and $X\to X'$, and using the facts that $Y$ is a homotopy colimit diagram and $i$ is a weak equivalence. Right? Feb15 comment Descent properties of spaces Yes, but the point is that it is not an abuse of language in the world of (infinity,1)-categories, because there is no issue of (co)fibrancy there. Feb15 comment Descent properties of spaces I imagine it is possible to translate everything there to model categorical language, though I can't say for sure as I haven't read the notes. Doing so would require being careful about taking (co)fibrant replacements and so on. If you are not familiar with (infinity,1)-categorical language, you can probably just read the paper keeping in mind that you should take (co)fibrant replacements whenever necessary. Feb15 comment Descent properties of spaces It looks like the author is implicitly using the language of (infinity,1)-categories. Feb12 revised Algebraic K-theory and Homotopy Sheaves added 2305 characters in body Feb12 comment Algebraic K-theory and Homotopy Sheaves @bananastack, in Thomason-Trobaugh this statement is Proposition 5.5.4. Feb12 revised Algebraic K-theory and Homotopy Sheaves added 2305 characters in body Feb11 revised Lifting DG-categories to characteristic zero added tag "dg-categories" Feb11 suggested approved edit on Lifting DG-categories to characteristic zero Feb10 comment When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories? @DavidWhite, the question of Hovey is about model structures on 2-categories, not about model structures on 2-relative categories. How is it relevant to this discussion? Feb10 answered When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories? Feb7 comment motivation of filtered colimits The term "compact object" surely deserves a place in this answer. Feb4 comment Fourier-Mukai transforms on stacks Feb4 comment When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories? Model categories enriched over SSet with the Quillen model structure are presentations of (infinity,1)-categories. Model categories enriched over SSet with the Joyal model structure are presentations of (infinity,2)-categories. See Remark 0.0.4 in [Jacob Lurie, (Infinity,2)-Categories and the Goodwillie Calculus I], arxiv.org/abs/0905.0462. Jan29 comment When is the cofibrant replacement of a product the product of the cofibrant replacements? This is true for coconnective dg-algebras, according to arxiv.org/abs/1112.2360. Jan29 answered An example of two cofibrant dg categories whose tensor product is not cofibrant Jan23 comment Connection between quasifibrations and homotopy cartesian squares Are you looking for something like this fibre-wise characterization: ncatlab.org/nlab/show/…? Jan21 awarded Curious