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For simplicial commutative rings this is proved in Lemma 3.1.2 of Schwede's “Spectra in model categories and applications to the algebraic cotangent complex”, and the proof there immediately extends …

One can make a reasonable claim that the analog of model categories in the realm of ∞-categories are model ∞-categories, see, for instance, http://arxiv.org/abs/1412.8411 and other papers by Aaron Maz …

If we write down the lifting square for an arbitrary cofibration $f\colon A→B$ and the unit map $η\colon X→RLX$ (with the bottom map being $b\colon B→RLX$), and then use the adjunction to pass to the …

This answer serves to record two explicit proofs of this fact in the literature: Corollary 5.1 in Raptis and Rosický, “The accessibility rank of weak equivalences”, arXiv:1403.3042v2. Theory and Appl …

The ∞-categorical limits (respectively colimits) are given by the right (respectively left) adjoint of the constant diagram functor $$C→C^I,$$ where $I$ is the indexing category and $C$ is the ∞-categ …

One can do even better than a Quillen adjunction: Theorem 3.8 in A model structure for Grothendieck fibrations establishes a Quillen equivalence between the projective model structure on presheaves of …

Since Goerss and Jardine do not give a (full) proof of this theorem, it is unclear how exactly this condition was intended to be used. However, this type of construction (where weak equivalences and f …

My question is: suppose that C is a cofibrant dg-category. Then are either of Ĉ or dgMod_C^op cofibrant dg-categories? A cofibrant object in a cofibrantly generated model category (such as dgCat) is …

No. For a counterexample to your claim, consider the model category M of simplicial presheaves on a small site S equipped with the projective model structure. Its fibrant objects are presheaves of Ka …

The counit map is cocontinuous in M, so using the fact that any cofibrant object is a retract of a transfinite composition of cobase changes of generating cofibrations of A-modules, combined with the …

For projective objects, see here: Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves. As explained there for presheaves of sets (and the same argument w …

There are many ways to define weak equivalences of simplicial sets without referring to topological spaces. A morphism f is a weak equivalence of simplicial sets if and only if one of the following e …

they seem to implicitly use the fact that (in their notation) if a map f is such that fa, fb, and fc are acyclic cofibrations, then ia(f) and ic(f) are again acyclic cofibrations. No, that's not …

Yes, this is always true. Replacing $X$ by its cofibrant replacement if necessary, we can assume $X$ to be cofibrant. In this case, $\def\Hom{\mathop{\rm Hom}} \Hom(X,-)\colon C→V$ is a right Quillen …

This depends on whether one insists on derived functors landing in the original model category (as is the case with modern approaches of Hinich and Dwyer–Hirschhorn–Kan–Smith), or in its homotopy cate …