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Let $\mathcal C$ be a cocomplete 1-category. Give it the model structure where every morphism is a cofibration and the weak equivalences are the isomorphisms. Then any (symmetric) monoidal closed stru …

Let’s prove that no such model structure exists, following Tom Goodwillie’s answer, and comments from Tom and from Tyrone. See Tom's comment for a simplified version of the following argument. In the …

There is Krause and Nikolaus' model structure on the category of group presentations whose homotopy category is the category of groups.

Let me see if I understand the question. If you start with one of the following pieces of data: a model category a relative category an $(\infty,1)$-category you can extract a homotopy category plus …

Zhen Lin points out below that I've been way too cavalier with transferring model structures along a reflection. So the following answer is not clearly correct. I will leave this up as community wiki …

In the years since this question was first asked, a new characterization of the fibrant objects in the injective model structure has been given by Mike Shulman in All (∞,1)-toposes have strict univale …

Here's half an answer. See Prop 5.2 of Dwyer and Kan's Function Complexes in Homotopical Algebra for a proof that the hammock localization of $C$ is the same as the hammock localization of the non-ful …

TL;DR : I'm not sure! (In the following, I basically elide the difference between (weak) factorization systems and classes of morphisms defined by (weak) lifting properties, which is often harmless, e …

If your goal is to understand how lifting properties in algebraic geometry fit into the bigger picture, I think the thing to say is that lifting properties are widely used in category theory for all s …

Not a complete answer. The following is inspired by Dmitri Pavlov's answer here. Let's recall the work of Dugger and Spivak. Let $\mathcal G$ be a "category of gadgets closed under wedges" (Definition …

Let $T$ be a well-founded poset and $k$ a field. Let $H: Ch_k \to Ho(Ch_k) = Gr_k$ be the homology functor and $\iota: Gr_k \to Ch_k$ be the canonical section which sends a graded vector space to the …

Here is a construction which I think is at least close to what you're driving at. Let $\mathcal S$ be our category of spaces, and let $Shv: \mathcal S \to Cat$ be the pseudofunctor taking a space to …

A "cleaner" form of the definition: Under Vopenka's principle the following are equivalent for a category $\mathcal C$: $\mathcal C$ is locally presentable; $\mathcal C$ is cocomplete, and equivalen …

Regarding Question 1, the only time I can think of when a Thomason fibrant replacement can be taken to be a 1-categorical localization is when $C$ has the homotopy type of the classifying space of a d …

EDIT: As Simon Henry points out in the comments, I've been too cavalier with accessibility issues. But since cofibrant generation of the flat maps is the only condition missing in the almost theorem b …