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I've encountered that condition a few time. Here is what I know about it: If that property is satisfied for a model category $M$ then the full subcategory $C$ of finite $I$-cell complex is itself a Br …

Here is a counter example in the general case: Take $E = [1] = 0 \to 1 $. $D = \{0\} \coprod \{1\} $ and $C = \{1\}$. where all maps are weak equivalence. The lax-pullback is $\{id:1 \to 1\}$, and the …

In short there isn't: the problem is that if you just have globular sets - and if you want $k$-cells to model $k$-arrows following the globular structure - then globular sets have no way of expressing …

Is there an argument in the litterature that show that every locally presentable $\infty$-category is equivalent to the localization of proper combinatorial Quillen model category ? Of course if one r …

In combinatorial and accessible weak model categories (also on ArXiv) I've studied Bousfield localization of weak and semi-model categories in both the combinatorial and accessible case. In particular …

This has not been done, and there are good reasons for it: While $sSet$-enriched categories are indeed very good to easily get examples of $\infty$-categories, they are very bad at understanding what …

I believe what you are after is the notion of "elegant Reedy category" This sort of things isn't true for a general Reedy category, but for an elegant one $R$ (see the link for the definition) if $\ma …

I am looking for an example where a transferred model structure fails to exist, even if one is willing to work with semi-model category. But let me be more precise: Let's say I have a combinatorial mo …

For a model category $C$, I'm denoting $h_\infty(C)$ the associated $\infty$-category (for example its Dwyer-Kan localization at weak equivalences, or if $C$ is simplicial the simplicial nerve of the …

It is not the case: the terminal semi-simplicial set $1$ is obviously fibrant but as I will show below the geometric product $1 \otimes 1$ is not fibrant. 1) What does $1 \otimes 1$ look like ? So, $1 …

Is the canonical (or Folk) model structure on the category of (strict) $\infty$-categories as constructed by Lafont, Métayer and Worytkiewicz in A folk model structure on omega-cat left proper ? All i …

As mentioned by David White in the comment, I've recently proved that left induced model structure exists (without any kind of large cardinal axiom) for any "tractable" class of cofibrations on a loc …

I'm looking for a Quillen model category which model the $2$-category of 'small category with finite limits (and functors between them preserving finite limits)': Left proper, right proper, Enriched …

Actually, you can, and you don't need accessibility (local presentability of the underlying category is enough). Under your assumption, for each $i:A \to B$ a generating cofibration, take $B \coprod_A …

Regarding (1) : A) Every model category has an associated $\infty$-category, obtained for example by taking the Dwyer-Kan localization at the class of all equivalence, (But there are other more expli …