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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 …

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' …

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 …

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 …

Note that category you've described is not locally presentable, but this is not a big deal -- if you use Delta-generated spaces (or some variant thereof) instead of general topological spaces, you're …

Yes, these agree. The usual model structures on $C = sSet$ and on $C = Top$ are both cartesian monoidal. So the functor $[-,X] : C^{op} \to C$ is a right Quillen functor when $X$ is fibrant (where $[ …

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 …

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 …

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 It's discussed in an unpublished note by Charles Rezk (either scroll down here to the note "A model category for categories", or …

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 …

I'll do you one better: you don't need a generating set of acyclic cofibrations. You just need what Simpson calls a pseudo-generating set, i.e. a set of acyclic cofibrations which suffices to detect f …

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 …

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 …