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Let $G$ be a discrete group. Consider the category of $G$-simplicial sets endowed with the injective model structure, i.e. cofibrations are the injective maps and weak equivalences are the maps which …

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 …

I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all t …

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 …

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 …

Let $\mathcal{T}= sh(C,J)$ a Grothendieck topos of sheaves over a (ordinary) site. One endows the category of simplicial presheaves over $C$ with the "Rezk-Lurie" model structure: that is we start wi …

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 …

Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ? It is well known to exists w …

Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one sit …

I'm looking for examples of proofs that some Quillen model categories are monoidal, or enriched over an other model category, which are based on explicit computation of the "pushout product" of the ge …

I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set. For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, …

I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those …

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 …