The n-Category Café

Combinatorial Model Categories

Posted by Urs Schreiber

Over the last years Jeff Smith has been thinking about – and notably talking about – but not publishing about – the idea and the theory of a certain type of model category called

combinatorial model category.

While he himself didn’t publish, the theory was found to be very useful and bits and pieces of it appear now more or less scattered in the works of other authors.

As far as I can tell the earliest published reference of the central recognition result for combinatorial model categories – now widely known as Jeff Smith’s theorem – is Tibor Beke’s Sheafifiable homotopy model categories.

Apart from helping find various new interesting model category structures, this and related (unpublished) results made Dan Dugger understand that the local projective model structures on simplicial presheaves which he had been studying exhaust precisely all (simplicially enriched) combinatorial model categories:

(Dan Dugger’s theorem) combinatorial model categories are precisely (up to Quillen equivalence) the

left Bousfield localizations

of the global projective model structures on functors $Func(C^{op},SSet)_{proj}$ from a small category $C$ to the category of simplicial sets.

I can’t know if it’s close to the way it happend, but if you read Jacob Lurie’s book Higher Topos Theory backwards, starting with the very last three propositions of the appendix, you can read it as the result of taking Dan Dugger’s theorem in Jeff Smith’s theory and finding its intrinsic model independent version:

Simplicial combinatorial model categories are precisely the models for locally presentable $(\infty,1)$-categories: those that are localizations – i.e. reflective $(\infty,1)$-subcategories – of $(\infty,1)$-categories of $(\infty,1)$-presheaves.

(Among these the left exact localizations are precisely the $\infty$-stack $(\infty,1)$-toposes.)

In total this provides us with the optimal situation where on the one hand we have a comprehensive abstract nonsense picture that tells us what is going on globally, while on the other hand we have highly developed concrete models and tools for realizing this abstract nonsense: the theory of combinatorial model categories.

Accordingly, we are eagerly awaiting Jeff Smith’s book to appear, whose upcoming existence keeps being hinted at in the literature. While that is not available yet, I thought it might be worthwhile to start compiling the available material in a useful coherent fashion at one place. First steps in this direction are at combinatorial model category and Bousfield localization of model categories.

One very useful input I already got on this was very kindly from Denis-Charles Cisinki, who pointed out the semi-published document by Clark Barwick: On left and right model categories and left and right Bousfield localization. Parts of that I have used in the construction of these entries. (All nonsense and other imperfections, of which there is still plenty, is my fault.) Much more needs to be done.