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In Abstract Homotopy Theory and Generalized Sheaf Cohomology Brown introduced the category of (fibrant) sheaves of spectra (he used Kan’s definition of spectra which is somewhat different from the definitions used nowadays) and proved that it is a category of fibrant objects. It is my understanding that model categories serving the same purpose have been constructed since then. I am interested in finding the original reference for such a model structure. I looked at some papers and nLab articles, but I’m unfamiliar with the jargon of the homotopy theory of sheaves and I have some trouble deciding whether the model structures I read about are the ones that I am looking for.
If yes:
The typical model category for sheaves of simplicial sets is usually called the Joyal-Jardine structure; it’s described in various papers of Jardine, and I believe it first appeared in print in one of those (perhaps around 1988 or so). It is consistent with a Brown-style fibrant objects structure on sheaves of simplicial sets (same homotopy theory). It’s worth pointing out that Brown’s “fibrant objects” (sometimes called “locally fibrant” in this setting) are not the same as the fibrant objects in the Joyal-Jardine model structure (“globally fibrant”). Globally fibrant implies locally fibrant, but not conversely.
Jardine has some nice recent papers about “cocycles”, which rehabilitate Brown’s point of view in a very elegant way.
For sheaves of spectra, Jardine has a number of papers on this too (I think he uses sheaves of symmetric spectra), though I have not looked closely at them.
In short: Jardine.
Perhaps you could also apply a general “stabilization” construction to sheaves of spaces?
I do feel obliged to point out that the Joyal-Jardine model structure on sheaves of spaces presents only the hypercompletion of the corresponding $(\infty,1)$-sheaf topos; for the full $(\infty,1)$-topos you seem to need to use simplicial presheaves.
Thanks for the answers!
The typical model category for sheaves of simplicial sets is usually called the Joyal-Jardine structure; it’s described in various papers of Jardine, and I believe it first appeared in print in one of those (perhaps around 1988 or so).
That’s probably Jardine, J. Simplicial presheaves (J. Pure Appl. Algebra 47 (1987), no. 1, 35–87). There is also a follow up paper Jardine, J. F. Stable homotopy theory of simplicial presheaves (Canad. J. Math. 39 (1987), no. 3, 733–747) which essentially carries out what Mike suggested. I think this is probably "the original reference" I was looking for.
Jardine has some nice recent papers about "cocycles", which rehabilitate Brown’s point of view in a very elegant way.
Indeed, in his recent papers Jardine uses "categories of cocycles", but I don’t see in what sense they "rehabilitate Brown’s point of view". Could you explain what you meant by that?
Maybe that’s too strong. But I’m thinking of this idea, which is one version of what Jardine does:
There’s a notion of “local fibration” of simplicial presheaves: a map which is a stalk-wise Kan fibration (the definition needs to be modified if your topos doesn’t have enough points). Likewise we have “local trivial fibrations”: stalkwise trivial Kan fibrations; these are called “hypercovers” by Brown.
If $X$ and $Y$ are simplicial presheaves which are locally fibrant (i.e., they are examples of Brown’s fibrant objects), then you can form a category $C(X,Y)$, whose objects are diagrams $(f,g)\colon E\rightarrow X\times Y$ (i.e., spans) such that $f$ is a hypercover, and whose morphisms are maps of spans. Then we obtain a simplicially enriched category: objects are locally fibrant simplicial presheaves, and maps from $X$ to $Y$ are the nerve of $C(X,Y)$. Composition is by usual composition of spans, using the fact that hypercovers are preserved under pullback. This simplicial category exactly models the Joyal-Jardine homotopy theory; the homotopy category is evidently the one that Brown constructs.
(Jardine doesn’t quite say all of this, but it’s not hard to read it off. He does give lots of nice little examples where you can compute things by directly examining the category $C(X,Y)$, or some variant of it.)
Maybe these references could be added to an appropriate nLab page.
Notice that the entry local fibration exists and says at least something in this direction.. Of course it could be expanded.
By the way, Jardine’s “cocycle category”-interpretation of K. Brown’s fibration categories can be refined to the full hom-homotopy types, pointers are at category of fibrant objects in this section here on derived hom spaces
But I thought Karol was asking about model structures on sheaves of spectra. This seems to be another issue…
Yes, that’s what I was talking about. Do we have a page about model structures on (pre)sheaves of spectra? Karol found a reference about that too.
Ah, I see. Yes, we have (brief) entries
By the way, Jardine’s “cocycle category”-interpretation of K. Brown’s fibration categories can be refined to the full hom-homotopy types,
I thought that’s what I said.
Maybe that’s too strong. But I’m thinking of this idea, which is one version of what Jardine does: (…)
Thanks for an explanation, I see what you mean. I find it very interesting since it seems that Jardine’s “rehabilitating Brown’s point of view” is closely related to rehabilitating I do in my own thesis. I will have to take a closer look at his examples.
For the record: I was asking for a model category with the same homotopy theory as the one defined by Brown. I didn’t put any specific requirements for its underlying category. In particular I’m equally happy with sheaves or presheaves, especially that Jardine shows that the sheafification functor is a Quillen equivalence between his model categories of simplicial sheaves and presheaves. I expect that this also holds stably.
@Charles: you did, right.
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