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added to combinatorial model category a reference (here) discussing that using a funny set-theoretic assumption about large cardinals, every cofibrantly generated mod cat is Quillen equivalent to a combinatorial one.
Weird that set-theory...
polished a bit and then created a separate section for Jeff Smith's theorem.
wanted to spell out the full proof there, but not yet
added to combinatorial model category the statement and detailed proof (here) that in a combinatorial model cat a kappa-filtered colimit over a sufficiently large kappa is already a homotopy colimit.
am pleased to be able to say that I typed into the entry combinatorial model category now the complete and detailed proof of Smith's theorem.
Well, there is one point that might need attention: in the first part of the proof, where one shows that these diagrams may be factored, somehow the argument the way I present it looks (while being close) a bit simpler than what I see in the literature. I expect, however, that this is only because I am missing some subtlety. But I leave it the way it is for the moment.
edited the list of basic examples slightly. Added the model structure for Cartesian fibrations as the $sSet_{Quillen}$-enriched replacement of the Joyal model structure
I started adding details on the proof of Dugger’s theorem
I have added to combinatorial model category, right after the proof of Smith’s theorem, two basic statemens useful for applying the theorem in the first place (that accessible full preimages of accessible full subcategories are accessible, and that the weak equivalences in a combinatorial model category form an accessibly included accessible full subcategory of the arrow category).
Also I have tried to stream-line the typesetting of the pointers to the references a bit more.
In combinatorial model category the line
This are corollaries 2.7 and 2..8 in Bar.
has a dead link. The reference to Barwick further down the page does not seem to correspond and Barwick’s homepage does not seem to have an article of that name. Does anyone have a live link for this?
The paragraph about Dugger’s Theorem under Characterization Theorems just said that all combinatorial simplicial model categories have presentations. Looking at Dugger’s paper, he actually shows that all combinatorial model categories have presentations. He gives a separate, easier proof in the simplicial case, though. I edited to reflect this.
Proof of Smith’s theorem: correct the argument for the solution set condition. 1. Adjust choice of J such that it contains the P->R constructed through factorization. 2. Clarify that only morphisms I->W are claimed to factor through W_0. Adjust W_0 to ensure this. 3. Factor P->Q instead of L+P->Q.
Gábor Braun
added pointer to:
added this pointer:
Added:
In the special case when weak equivalences are closed under filtered colimits and the model structure is left proper, the statement of Smith’s theorem can be simplified.
The following formulation can be found as Proposition A.2.6.15 in Lurie \cite{Lurie}.
\begin{theorem} Suppose $C$ is a locally presentable category equipped with a class $W$ of weak equivalences that turn it into a relative category and a set $I$ of generating cofibrations. If the following conditions are satisfied, then $C$ admits a left proper model structure with $I$ as the set of generating cofibrations:
$inj(I) \subset W$;
every element of $I$ is an h-cofibration;
the class $W$ is perfect: it satisfies the 2-out-of-3 property, is closed under filtered colimits (in the category of morphisms in $C$), and is generated under filtered colimits by a set of morphisms. \end{theorem}
[ sorry, wrong thread – removed ]
I have moved up Dugger 2001 in the list of references, and adjusted the wording around it, in order to clarify that, apart from giving “Dugger’s theorem”, this article is (in its Section 2) an early explicit account of the notion of combinatorial model categories
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