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Here is an example that surprised me at some time in the past. Bisson and Tsemo introduce a nontrivial model structure on the topos of directed graphs. Here a directed graph is simply a $4$-tuple $(V, …

Here is what André Joyal wrote in an email to me: No, I have not discovered the model structure for quasi-categories in the 1980's. I became interested in quasi-categories (without the name) around 1 …

Model categories provide a powerful framework for commuting (homotopy) limits and colimits, and, more generally, for commuting left adjoint functors and (homotopy) limits, as well as right adjoint fun …

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 …

if we don't assume properness, I don't even see why the first is homotopy-invariant! The pushout of a diagram A←B→C in which all objects are cofibrant and one of the maps is a cofibration is always …

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 …

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 …

On the other hand, the category of commutative monoids seems to be much more subtle. The conditions for the existence of a model structure on commutative monoids were worked out by Jacob Lurie aro …

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 …

Yes, it is also available here: https://www.math.uni-bielefeld.de/sfb343/preprints/pr97044.ps.gz

if a simplicial presheaf F:Man^op→sSet satisfies homotopy descent, where Man is the category of smooth manifolds, then there exists a K such that F≅[−,K]. Here one must also mention that F is requir …

Is it true that the classification can be recovered from the ho(ChQ)? Yes, the canonical functor Ho(Fun(T,M))→Fun(T,Ho(M)) induces a bijection on isomorphism classes if T is a tree and M is the rela …

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 …

Yes, this is proved in https://arxiv.org/abs/1506.01475 by Thomas Nikolaus and Steffen Sagave. If you need a specific monoidal left Quillen equivalence, you can upgrade Dugger's result to the symmetr …

More precisely, let A,B be two simplicially enriched model categories, is it true that every adjoint pair of ∞-functors between A,B comes from a Quillen pair? Assuming the model categories are combi …