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If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial sets. These are called the injective and projective model structures, and in both the weak equivalences are the levelwise weak equivalences, i.e. those natural transformations $X \to Y$ such that $X(c) \to Y(c)$ is a weak equivalence for all $c \in C$.

The cofibrations in the injective model structure are the levelwise cofibrations. The injective fibrations are then defined as those maps which have the left-lifting property with respect to all cofibrations which are also weak equivalences. The injective fibrant objects are those simplicial presheaves in which the map to the terminal presheaf is an injective fibration.

My Question: Is there a better characterization of the fibrant objects? If I have an object are there any shortcuts which I can use to test if it is fibrant? An obvious necessary condition is that each simplicial set $X(c)$ must be fibrant (i.e. a Kan complex), but I want sufficient conditions. I'm willing to put restrictions on C.

Dually, we can look at the projective model structure where the fibrations are the levelwise fibrations, and cofibrations have the extension property with respect to these. I'd also be interested in knowing about the cofibrant objects in this model structure, although for now I'm more interested in the injective model structure.

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I know that the identity functor is a Quillen equivalence between these two model structures, and when C is Reedy the identity functor is also a Quillen equivalence with the Reedy model structure. So what I'm asking is definitely specific to the injective model structure. I don't want to replace it with an equivalent one. –  Chris Schommer-Pries Dec 21 '09 at 21:23
    
Meanwhile Richard Garner gave a nice characterization of cofibrant simplicial presheaves in the projective structure, which answers the question asked in the last paragraph: mathoverflow.net/questions/97690 –  Dmitri Pavlov May 1 '13 at 19:52
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3 Answers

up vote 5 down vote accepted

In the introduction to his paper "Flasque Model Structures for Presheaves" (in fact simplicial presheaves) Isaksen states on the top of page 2 that his model structure has a nice characterisation of fibrant objects and that "This is entirely unlike the injective model structures, where there is no explicit description of the fibrant objects". This would answer your question. It might be my ignorance, but I think there is no justification for Isaksen quoted statement except that no characterisation is known as yet.

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Well, it would answer the first part of my question, but maybe there are still some sufficient conditions which guarantee that a given simplicial sheaf is fibrant, no? There seem to be lots of necessary conditions floating around but not so many examples or sufficient conditions. –  Chris Schommer-Pries Dec 21 '09 at 21:10
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This is the answer I agree with the most. There just is not going to be a good answer to this question in general, I don't think. You are going to need specific properties of your category C. Just think; you need the right lifting property with respect to every levelwise trivial cofibration A --> B of diagrams. For this to work, you are going to need some characterization of such levelwise trivial cofibrations, along the lines of Baer's criterion for injectivity of modules over a ring. –  Mark Hovey Dec 22 '09 at 1:01
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I'm not 100% sure, but I think the answer is that you should choose a cellular model for PSh(C) (the category of presheaves of sets on C), which is a set S of monomorphisms in PSh(C) such that every monomorphism of PSh(C) can be written as a transfinite composition of pushouts of elements of S. Then the fibrant objects in the category of simplicial presheaves on C are those objects X such that X(b) → X(a) is a fibration for every a → b in S (here X(a) denotes the simplicial set whose n-simplices are HomPSh(S)(a, Xn)).

Finding a cellular model is not so easy in general (see Lemma A.3.3.3 of Jacob's book for a proof of existence). But in special cases it's simple, e.g., for C = Δ, we may take the maps $\partial \Delta^n \to \Delta^n$ for $n \ge 0$.

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Can you elaborate on this? I think I can find the subset S which "generates" the monomorphism in the cases I'm thinking about (which you can probably guess, Reid). Do you know any references for this besides HTT? –  Chris Schommer-Pries Dec 21 '09 at 21:16
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Well, you can look in Cisinski's thesis Les préfaisceaux comme modèle des types d'homotopie available at his web site www-math.univ-paris13.fr/~cisinski/publications.html for the terminology (Definition 1.2.26). By Lemma 2.3.2 what I wrote in the first paragraph is true if you replace fibration by acyclic fibration everywhere. But I don't really know whether it's true as written in general. –  Reid Barton Dec 21 '09 at 22:18
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What you need is a cellular model of the levelwise trivial cofibrations, not a cellular model of the monomorphisms. This is in general very difficult, and depends heavily on your category C. –  Mark Hovey Dec 22 '09 at 0:58
    
OK, I was worried that might be the case. I was hoping I could get such a cellular model by forming the pushout product of a cellular model for C with the set of horn inclusions. That seems to work at least some of the time. Do you have an example where it fails? –  Reid Barton Dec 22 '09 at 1:20
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@philip314: I was replying to Reid's use of the term "cellular". It is true that this is new usage to me, but maybe Jacob Lurie introduced it. @reid: Take any Bousfield localization; the generating cofibrations are the same, but the generating trivial cofibrations get completely out of control in general. You need a cardinality argument to get them. –  Mark Hovey Dec 22 '09 at 21:03
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I think you can find an answer in my question about global fibrations of simplicial sheaves: global fibrations of simplicial sheaves .

There, Andreas Holmstrom pointed me to Voevodsky's preprint "Homotopy theory of simplicial presheaves in completely decomposable topologies", http://front.math.ucdavis.edu/0805.4578 , where I discovered lemma 4.1 , which I think aswers your question.

Although it is not proved, Voevodsky says that it is straightforward. I could manage myself to write a proof of it, at least under the hypothesis of Brown-Gernstern's "Algebraic K-theory as generalized sheaf cohomology", in LNM 341/1973, that is, with sheaves defined on a Noetherian space of finite Krull dimension.

In this situation at least, fibrations are global fibrations. That is, a morphism of sheaves $p: E \longrightarrow B$ is a global fibration if and only if 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.

As a corollary, if you take $B = *$, this condition tells you that fibrant objects are those for which each restriction map $E(V) \longrightarrow E(U)$ is a Kan fibration. In particular, put $U=\emptyset$ and this implies that each $E(V)$ must be a Kan complex.

All this with Brown-Gersten's hypothesis, but Voevodsky doesn't seem to need them, so maybe it is also true in your situation.

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So I had a brief look at the Voevodsky preprint. Right before Lemma 4.1 he introduces a class of morphisms "J_P" which is roughly the class of representable presheaves times a horn-inclusion plus a relative version of these for "distinguished squares". Then the Brown-Gernstern fibrations are those which have the right lifting property w.r.t. the maps in J_P. Unless I'm missing something this is not the same as an injective fibration, which would have the lifting property w.r.t. all levelwise cofibrations. What are the B-G cofibrations? –  Chris Schommer-Pries Dec 21 '09 at 21:06
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@Chris. I think this is not a major problem: J_P is (I think) a class of generating (trivial) cofibrations. In a cofibrantly generated model category structure is enough to have the RLP with respect some (trivial) cofibrations to have it with respect all of them. The problem with my answer, I believe, is that I didn't realize your weak equivalences are not those of B-G: yours are pre-sheaf weak equivalences, "open-wise" defined, theirs are sheaf weak equivalences, that is stalk-wise. Sorry for my mistake. –  a.r. Dec 22 '09 at 3:56
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