SHEAFIFICATION AS FIBRANT REPLACEMENT

H. FAUSK

1. introduction

We give a model structure on the category of presheaves of sets on a Grothendieck site. The weak equivalences are maps whose associated map of sheaves is an isomorphism. The fibrant replacement functor is given by sheafification.

This model structure is a sidetrack from work on a model structure on the category of cochain complexes of (pre)sheaves whose homotopy category is the usual derived category [3]. The motivation for writing this note is pedagogical; we provide a simple, yet not entirely trivial, example of a model structure.

2. The model structure on presheaves

Let C be a small Grothendieck site. Let E denote the topos of sheaves of sets on C, and let P denote the topos of presheaves on C, i.e. the category of sheaves of sets on the chaotic topology on C. The original source for topos theory is [1].

There is a sheafification functor L^{2} : P → E [1, II.3.0.5,3.2]. L^{2} functor is left adjoint to the inclusion functor i: E → P. These two functors, together with an explicite adjunction, consititute a morphism of topoi E → P.

Proposition 2.1. There is a model structure on P such that

(1) the cofibrations, C, are all maps in P

(2) the weak equivalences, W, are maps f such that L^{2}f is an isomorphism

(3) the fibrations, F, are retracts of pullbacks of maps of the form L^{2}(f) for f ∈ P.

Proof. The category P is complete and cocomplete. The class W satisfies the two-out-of-three axiom, and C, W, F are all closed under composition and retract.

We show that inj (W) ∩ W is the class of all isomorphisms. The intersection inj (W) ∩ W contains all isomorphisms. Conversely, let f: X → Y be both a fibration and a weak equivalence. Then the diagram

Date: March. 2. 2006.

[figure: X; f; X; f; Y; Y]

has the lifting property. Hence f is an isomorphism.

A factorization of f: X → Y as X →^{f} Y →^{=} Y is a factorization of f as a cofibration followed by an acyclic fibration.

We give a factorization of any map as an acyclic cofibration followed by a fibration. Let f: X → Y be an arbitrary map. We form a factorization of f as follows:

(2.2)  \xymatrix{X \ar[r] \ar[dr]_{f} & Z \ar[r] \ar[d] & L^{2}X \ar[d]^{L^{2}f} \\  & Y \ar[r]^{η(Y)} & L^{2}Y}

where Z is the pullback of L^{2}f and η(Y).

The map η(Y) is an L^{2}-isomorphism, and the functor L^{2} respects finite limits. Hence L^{2}Z → L^{2}X ≅ L^{2}L^{2}X is an isomorphism. By the two out of three property for W the map X → Z is in W. We need to show that Z → Y has the right lifting property with respect to all maps in W. Let

(2.3)  \xymatrix{A \ar[r] \ar[d]_{g} & Z \ar[d] \\ B \ar[r] & Y}

be a diagram where g ∈ W. Apply the functor L^{2} to diagram 2.3 gives a new diagram which has a lift h: L^{2}B → L^{2}Z since L^{2}g is an isomorphism. Since Z is the pullback in 2.10, a diagram chase shows that h gives rise to a lift h′: B → Z in diagram 2.3. The retract argument gives that fibrations are retracts of pullbacks of maps between sheaves.

The retract argument and the factorization of a map as an acyclic cofibration followed by a fibration give that W = proj inj W. □

Proposition 2.4. The fibrant objects in P are precisely the sheaves. A functorial fibrant replacement functor is given by sheafification η: 1 → L^{2}.

Give E the model structure where the weak equivalences are the isomorphisms.

Proposition 2.5. The map of topos E → P is a Quillen equivalence.

Hence the homotopy category of the model structure on P is equivalent to the category E of sheaves.

Remark 2.6. It is not clear if the model structure on P is cofibrantly generated. There is a set of maps generating the class of all isomorphisms of presheaves. But it is not clear if there is a set of maps generating the fibrations. For any C ∈ C let rC denote the presheaf represented by C. A map f: X → Y is a (levelwise) injective map of presheaves if and only if it has the right lifting property with respect to all suspensions of maps of the form rC ∐ rC → rC, for C ∈ C, where the map is the folding map (given by the identity maps on C). A map f: X → Y is a (levelwise) surjective map of presheaves if and only if it has the right lifting property with respect to all suspensions of maps of the form ∅ → rC, for C ∈ C.

One idea is to use the pro-category of presheaves, where the stalk functor can be represented.

We formalize the method we used to give the model structure on P. This gives a way to produce model structures that does not rely on the small object argument. We first give an immediate extension of Proposition 2.1 and then give a more general result with a model structure as starting point.

Let C be a category and let T: C → C be a functor.

Definition 2.7. We say that a map f: X → Y is an T -equivalence if T (f): T (X) → T (Y) is an isomorphism. Denote the class of T -equivalences by W_{T}.

We assume that T is a localization. That is, T is left adjoint to the inclusion functor i: T C → C and T η = ηT is a natural isomorphism. Here η: 1 → iT is the counit natural transformation of the adjunction.

Lemma 2.8. Assume the left adjoint functor T: C → T C respects pullback diagrams of the form

(2.9)  \xymatrix{Z \ar[r] \ar[d] & T X \ar[d]^{T f} \\ Y \ar[r]^{η(Y)} & T Y.}

Then there is a model structure on C with cofibrations all maps in C, weak equivalences W_{T}, and fibrations inj (W_{T}). Moreover, the functorial factorizations of a map f: X → Y are given by

X →^{f} Y = Y

and by the pullback diagram

(2.10)  \xymatrix{X \ar[r] \ar[dr]_{f} & Z \ar[r] \ar[d] & T X \ar[d]^{T f} \\  & Y \ar[r]^{η(Y)} & T Y}

Lemma 2.11. Let C be a model structure with C, W, F the classes of cofibrations, weak equivalences, and fibrations, respectively. Let \overline{W} be a class of maps containing W. Assume that \overline{W} is closed under composition and retract, and that it satisfies the two out of three property. Let η: 1 → T be a natural transformation between functors C → C. Assume the following conditions:

(1) η(X): X → T X is in \overline{W} for all X ∈ C

(2) if f: X → Y is in \overline{W} ∩ C, then T (f) is in W ∩ C

(3) if f: X → Y is in F, then T (f) is in F

(4) the pullback of a map in \overline{W} along a fibration is again in \overline{W}

(5) the two natural transformations ηT, T η: T → T^{2} agree and are both

isomorphisms. Then there is a model structure on C such that

(1) the cofibrations are C

(2) the weak equivalences are \overline{W}

(3) the fibrations are inj (\overline{W} ∩ C)

We omit the easy proof of Lemma 2.11. The conditions in the Lemma are not optimal. There is room for improvements.

Proposition 2.1 can be extended to presheaves of O-modules for a sheaf of rings O in E. Let M be the category of O-modules in E, and let N be the category of presheaves of i^{∗}O-modules in P. Let M have the model structure where the weak equivalences are the isomorphisms. One can describe a model structure on N and a Quillen equivalence M → N as before. This also follows directly from Lemma 2.11.

The technique used to construct model categories used in this note might be related to [2, App.]. (I have not been able to check the reference.)

References

[1] M. Artin, A. Grothendieck, and J. L. Verdier, Thorie des topos et cohomologie étale des schémas, Tome 1: Théorie des topos. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Mathematics, Vol. 269. Springer-Verlag, Berlin-New York, 1972.

[2] A.K. Bousfield, E.M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, pp. 80–130, Lecture Notes in Math., 658, Springer, Berlin, 1978.

[3] H. Fausk, The projective model structure on chain complexes of presheaves. Preprint 2006.

   E-mail address: fausk@math.uio.no