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I think my preprint answers this question in the appendices when I compare the horizontal Joyal model structure on $\Theta[C]$ sets with the Rezk model model structure for $\operatorname{Se}_C\cup \op …

The condition you've stated implies that the homotopy sheaves are equivalent, and it is implied by the map being a weak equivalence, so they are equivalent. You're nullifying $\mathbb{A}^1$ in the $\i …

It's actually something you already know: It is the fibrewise groupoidification of the free cartesian fibration. The free cartssian fibration functor sends a functor $$p:A\to B\mapsto p': A\downarrow …

A recipe for a counterexample: Let $(X,e:\Delta^0\to X,m:X\times X\to X)$ be monoid object in the homotopy category of spaces $h\mathcal{S}$ (that is, an H-monoid). Note that this is a property of t …

In the language of $\infty$-categories, which makes it a bit clearer, this is asking for the reflector (left adjoint) of the inclusion of a reflective subcategory to preserve filtered limits. This is …

First, notice that if $X\hookrightarrow Y$ is an injective map over $S$, then the map $M_{X,\phi} \to M_{Y,\phi}$ is a cofibration of simplicial categories. To see this, notice that it is a pushout o …

The full subcategory spanned by the fibrant objects of a model category always satisfies the right Ore condition, following https://ncatlab.org/nlab/show/calculus+of+fractions . Similarly the subca …

First question: I couldn't find anything. Second question: I just found a sufficient condition under some strong finiteness assumptions: According to the paper of Dundas, Röndigs, and Østvær, the enr …

No, the most we can say is that there exists a zig-zag. $a\leftarrow Qa\rightarrow U(b)$ where the first arrow is the the component of the natural weak equivalence $Q\to Id$ with $Q$ the cofibrant r …

The trick is to check that the corner map $$\lambda^n_k\bar{\times}\delta^m:\Lambda^n_k \times \Delta^m \coprod_{\Lambda^n_k\times \partial \Delta^m} \Delta^n \times \partial \Delta^m \hookrightarrow …

Edit: It appears that this is wrong! See the comments below. Using Finn's observation that it is the subobject classifier, we can see that it is the nerve of the contractible groupoid with exactly t …

The condition you're looking for is called combinatoriality (and local presentability). A model category is combinatorial provided it satisfies some complicated conditions involving accessibility, bu …

Dear Saul, The answer to your question is the subject of chapters 16-19 of Phil Hirschhorn's book Model Categories and their Localizations. To write out the answer in the general case would be pro …

It always has a model structure using Kan's theory of Reedy categories. For a proof, see Hirschhorn Model Categories and their Localizations 15.3. This is because $\Delta$ and $\Delta^{op}$ are both …

You must allow the weak equivalences to be unpointed homotopy equivalences. These become honest pointed homotopy equivalences between fibrant-cofibrant objects by the generalized whitehead theorem. S …