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Browsing through the old unanswered questions, I've come across this one, which happily can be partially answered now by the work of Riehl and Verity (Zhen Lin will be aware of this, which is why I'll …

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 …

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 …

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 …

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 …

My go-to reference for inducing model structures along an adjunction is Hess, Kedziorek, Riehl, and Shipley's A necessary and sufficient condition for induced model structures, which works in great ge …

Yes, the canonical model structure is unique. The uniqueness of the canonical model structure on $Cat$ was nicely exposited by Chris Schommer-Pries on the Secret Blogging Seminar back in the day. Let' …

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 …

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.

I asked Chris Schommer-Pries (the author of the blog post) about this. It's discussed in an unpublished note by Charles Rezk (either scroll down here to the note "A model category for categories", or …

Nikolaus has shown (see Cor 2.21) that every combinatorial model category where all trivial cofibrations are monic is Quillen equivalent to its category of algebraically-fibrant objects, in which ever …

Lurie uses A.2.6.7 to prove the "easy" direction of Jeff Smith's theorem in A.2.6.8, namely that every combinatorial model category arises from the construction of the theorem. This part of the theore …

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 …