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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 …

It's been more than a year and a half since I asked this question and I had a lot of thought about it so I decided I will post my own answer. First I entirely agree with Yonatan that the main problem …

In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity categori …

For this to work, it is best to identify connective spectrum with spaces equipped with a group-like $E_\infty$-algebra structure (these are equivalent). From this point of view: $\mathbb{Z}$ is the f …

While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems like somet …

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 …

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 …

Assuming you work with unpointed spaces (but the example can easily be adapted to the pointed case) the map $1 \to 2$ gives a counterexample : its fiber are $1$ and $\varnothing$ so they are both fini …

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 …

As I said in the comment, this would involve a very large number of different references! (almost one by subsection...) But to some extent, contributors to the nLab already started doing that and it i …

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 …

I fix $C$ a symmetric monoidal model category (with a cofibrant unit if it helps). I'm assuming that it is closed, or at least that the tensor product commutes to colimits in each variable. If $X$ is …

I know from the the theory of Artin groups that, as the $K(\pi,1)$ conjecture is known for Braids group, that using Salvetti complexes we have a fairly explicit finite CW-complex presentation of the c …

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 …