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It is true. You can reduce to the case of simplicial sets by using the fact that $\mathbf{R} \mathrm{Hom} (T, -)$ preserves homotopy limits and (allowing $T$ to vary) is jointly conservative. Finding …

There is no difference for $\kappa = \aleph_0$. The point is that you can build colimits for filtered diagrams using just colimits for chains. Every filtered category $\mathcal{J}$ admits a cofinal …

If by "additive" you mean an $\mathbf{Ab}$-enriched category with a zero object and biproducts, then yes. Let $\mathcal{M}$ be model category that is additive in this sense, let $\mathcal{M}_c$ be the …

One of Quillen's original examples was the category of chain complexes of finitely-generated modules over a ring – this is obviously equivalent to a small category, and of course, one has to use Quill …

It seems the answer is indeed yes in great generality – certainly at least for any complete wellpowered category. By a result of Cassidy, Hébert, and Kelly [Reflective subcategories, localizations and …

Let $\mathcal{M}$ be a category, let $\mathcal{M}^\circ$ be a reflective (full) subcategory of $\mathcal{M}$, and let $L : \mathcal{M} \to \mathcal{M}^\circ$ be the reflector. Question. Does there exi …

There are only nine possible model structures on $\mathbf{Set}$. You can easily check that none of them give the right weak equivalences: Either everything is a weak equivalence, or $X \to Y$ is a w …

It's not quite in the literature, but there is a fully explicit construction that avoids hammock localisation or any kind of fibrant replacement: by a recent result of Lennart Meier, a certain "double …

Yes, $D_0 \times_{D_1} D_2 \to (E_2 \times_{E_1} E_0) \times_{E_0} D_0$ is a fibration. First, observe that $$(E_2 \times_{E_1} E_0) \times_{E_0} D_0 \cong E_2 \times_{E_1} D_0 \cong (E_2 \times_{E_1} …

Yes, $\wedge$ is invariant under weak homotopy equivalence in $\mathbf{sSet}_*$. It suffices to show that it sends anodyne extensions (= trivial cofibrations) to weak homtoopy equivalences: indeed, ev …

Here is a somewhat degenerate example that illustrates what can go wrong. Let $\textbf{Ab}$ be the category of abelian groups, considered as a homotopical category where the weak equivalences are the …

Let $U$ and $V$ be Segal spaces and let $f : U \to V$ be a Dwyer–Kan equivalence. That means two things: The following diagram is a homotopy pullback square: $$\require{AMScd} \begin{CD} U_1 @>>> U_0 …

As pointed out, the answer is no – because we cannot hope to have fibrant replacements in general. On the other hand, by following the programme of van Osdol [1977, Simplicial homotopy in an exact cat …

In [Schwänzl and Vogt, Strong cofibrations and fibrations in enriched categories], the authors refer to an earlier preprint, [Schwänzl and Vogt, Cofibration and fibration structures in enriched catego …

I work in a general pointed model category. The homotopy fibre of a morphism $f : Y \to X$ can be defined as follows: first, choose a fibrant replacement $w_X : X \to \hat{X}$ and then factor $w_X \ci …