What are the fibrant objects in the injective model structure? - MathOverflow

What are the fibrant objects in the injective model structure?

2009-12-21T17:08:09Z

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.

Answer by Reid Barton for What are the fibrant objects in the injective model structure?

2009-12-21T18:15:25Z

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 Hom_PSh(S)(a, X_n)).

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 $n \ge 0$.

Answer by Agusti Roig for What are the fibrant objects in the injective model structure?

2009-12-21T20:09:35Z

I think you can find an answer in my question about global fibrations of simplicial sheaves: http://mathoverflow.net/questions/5179/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.

Answer by philip314 for What are the fibrant objects in the injective model structure?

2009-12-21T20:11:25Z

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.