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I'm reading the classical Brown-Gersten's paper "Algebraic K-theory as generalized sheaf cohomology" and I'm stuck with their choose of global fibrations. Namely, a morphism of simplicial sheaves $p : E \longrightarrow B$ is a global fibration if for every inclusion of open sets $U\subset V$ the natural map $E(V) \longrightarrow B(V) \times_{B(U)} E(U)$ is a (Kan) fibration of simplicial sets.

My problem is: why these fibrations? As far as I can see, when they make use of this definition in constructing the factorizations of the model category structure, they could have chosen the fibrations to be defined open-wise: $p : E \longrightarrow B$ is a fibration if $p(V) : E(V) \longrightarrow B(V)$ is a (Kan) fibration of simplicial sets for every open set $V$ and apply as well the small object argument they use at this point.

In other contexts I understand this kind of fibrations. For instance, for the model structure of the category of diagrams $C^I$ of a model category $C$ when $I$ is a 'very small' category (Dwyer-Spalinski, "Homotopy theories"), or a Reedy category. In these cases, this kind of fibrations ensures that you can extend your liftings by induction. But I don't see if this is their role with a category of sheaves, since no induction seems to be at hand.

A colleague of mine has said to me thas this choice of fibrations is the consequence of choosing the cofibrations to be the monomorphism, following Joyal's "Letter to Grothendieck"; that is, these are precisely the fibrations if you choose monomorphisms as cofibrations and ask fibrations to have the RLP with respect to trivial cofibrations. But I couldn't find anywhere this famous Joyal's letter, so I would also be glad if someone could tell me where I can find it.

Thanks in advance for any hints.

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The part about where Joyal's letter can be found has now been answered here. –  David Richerby Dec 19 at 15:41
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Thank you for remembering this question of mine in relation with Joyal's letter, David: I really appreciate it. –  a.r. Dec 20 at 15:51

2 Answers 2

up vote 5 down vote accepted

For model structures on simplicial sheaves, there is a difference between the Joyal-Jardine approach and the Brown-Gersten approach. This is well explained in Voevodsky's preprint: Homotopy theory of simplicial presheaves in completely decomposable topologies, available here. Briefly, the Brown-Gersten approach does not work for arbitrary sites, but it works for a class of sites defined in Voevodsky's paper - this class includes Noetherian finite-dimensional spaces. When the B-G approach works, the resulting model structure has better finiteness properties than the Joyal-Jardine model structure, which on the other hand can be defined for simplicial (pre)sheaves on any site.

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Ok. It seems to me that the result I was looking for is Voevodsky's lemma 4.1 in your reference, that for topological spaces says that a map of sheaves p: E ---> B has the RLP with respect to trivial inclusions if and only if it verifies Brown-Gersten's definition of fibration (the one I put in my question and, shame on me, Voevodsky says that this is trivial). So, Brown-Gernsten's model structure should be an injective one (cofibrations defined open-wise). Thank you both, Urs and Andreas, for your hints. –  a.r. Nov 16 '09 at 17:17
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Strictly speaking, the Brown-Gersten-Voevodsky model structure does make sense for arbitrary sites, as explained by Daniel Isaksen in his paper “Flasque model structures for presheaves”. What does not generalize to arbitrary sites is the description of the sheaf condition in terms of Nisnevich squares. –  Dmitri Pavlov May 3 '13 at 14:14

Here a quick reply, as I have to rush, not yet directly answering your question:

all there is to the various model structures on simplicial (pre)sheaves -- those where cofibrations are objectwise injections (the injective ones) and those where the global fibrations are the objectwise Kan fibrations (the projective ones), global and local, on sheaves and on presheaves -- I have tried to collect here:

model structure on simplicial presheaves

When I wrote this -- as you can see from the big diagram there -- I seem to have been under the impression that what Brown-Gersten describe is the local projective model structure, as afterwards described in much detail by Daniel Dugger and collaborators (see the references listed there).

But I never looked much at Brown-Gersten, so now before answering your question definitely I would want to go back and check their article.

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Thank you. I'll take a look at your references. –  a.r. Nov 15 '09 at 20:10
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Daniel Isaksen in his paper “Flasque model structures for presheaves” generalizes the Brown-Gersten model structure to arbitrary sites. In particular, he removes the Noetherian condition on the Grothendieck topology. The resulting flasque model structure lies between the projective and injective structures. The work of Brown-Gersten and the more recent paper by Voevodsky can then be interpreted as providing a simple condition for fibrancy in the flasque model structure when the Grothendieck topology satisfies some conditions. –  Dmitri Pavlov May 2 '13 at 13:28

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