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Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which o …

Is there a fibrant replacement functor in the Joyal model structure which can be described non-recursively, like $Ex^\infty$ for the Quillen model structure? I believe another way to put this is to as …

Let $Q$ be the homotopical category of combinatorial model categories and left Quillen functors, with left Quillen equivalences for weak equivalences. Let $\mathbf Q$ be the corresponding $\infty$-ca …

I'm looking for a proof of the existence of the Joyal model structure -- with its usual description -- which uses Cisinski theory directly. The closest thing I know of is Theorem 5.26 of Ara's Higher …

Let $C$ be a small category and $M$ a model category. Then there are various "global" model structures (projective, injective, Reedy) on the category $Fun(C,M)$ of functors from $C$ to $M$, all with t …

Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a …

In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy t …

How explicit are the model structures for various categories of spectra? Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit generat …

If $C$ is a cofibrantly generated model category which is also monoidal biclosed, then to check that $C$ is a monoidal model category, it suffices to check that the Leibniz product of generating cofib …

The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions. Thi …

Let $\mathcal{C}$ be a cofibrantly-generated model category. My impression is that the following two conditions are highly correlated: $\mathcal{C}$ is right proper. There is an explicitly-describab …

The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations? More generally, let $C$ be a locally presentable cat …

In a model category, I have tools to show that mapping spaces are contractible. But if I want to show a mapping space is empty or contractible, is there anything I can do on general grounds? The idea …

This is true for orthogonality classes- see Corollary 6.24 in Adamek and Rosicky - but I can't seem to find this result in the literature for weak orthogonality. Here, by a weak orthogonality class i …

I have algebraic models of $n$-categories in mind. By "theory of (weak) $n$-categories", I mean "[monad / operad / whatever] whose algebras are (weak) $n$-categories". To be more precise: fix an alge …