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local model structure on simplicial presheaves

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Idea

See model structure on simplicial presheaves.

Definition

There are many model category structures on the category of simplicial presheaves derived from the model structure on simplicial sets.

The local such model structures are of interest in that they model infinity-stacks so that they are a presentation of the (infinity,1)-category of (infinity,1)-sheaves on the given site.

They can be thought of as being obtained from global model structures, of which there are two:

These two model structures are Quillen equivalent (DHI04 p. 5 with the Quillen equivalence given by the identity functor). They can be defined on any domain category SS, not necessarily a site. If we do have a structure of a site on SS then there is a notion of local weak equivalences of simplicial presheaves on SS, defined below. One gets local projective and local injective model structures by applying left Bousfield localization of the above model structures at local weak equivalences (see p. 6 of DHI04)

  • the local projective model structure (weak equivalences are locally (usually stalkwise) and cofibrations are those that have the left lifting property against objectwise acyclic fibrations);

  • the local injective (weak equivalences are locally (usually stalkwise) and fibrations are those that have the right lifting property against the objectwise acyclic cofibrations).

Warning Since the (homotopy classes) of weak equivalences do not form a small set, the general existence theorem recalled at Bousfield localization of model categories does not apply. The existence of the Bousfield localization has to be shown by hand. For the injective structure this is what Joyal and Jardine accomplished.

Again, the injective and projective local model structures are Quillen equivalent by the identity functors between the underlying categories and hence provide projective and injective versions of the corresponding homotopy theory of infinity-stacks.

In the local injective structure all objects are cofibrant, so that the opposite category of simplicial presheaves with the local injective model structure is a category of fibrant objects.

Both local model structures are proper simplicially enriched categories (DHI04 p. 5).

The local injective model structure on simplicial presheaves is originally due to Jardine, following the construction of the Quillen equivalent local model structure on simplicial sheaves by Joyal. It was only later realized in DHI04 as a left Bousfield localization of the global injective model structure.

In between the injective and the projective model structures there are many other model structures obtained by varying the class of generating global cofibrations.

In the following let SS be a small site and denote by SimpPr(S)SimpPr(S) be the category of simplicial presheaves on SS.

Local model structures

One usually says that a local model structure on a category of presheaves is one whose weak equivalences are not defined objectwise but on covers and/or on stalks.

Local weak equivalences

There are different equivalent ways to define local weak equivalences of simplicial presheaves on a site SS.

In terms of sheaves of homotopy groups

(see section 2 of Jardine07)

We want to say that a local weak equivalence of simplicial presheaves is one which is “over each point” an isomorphism of homotopy groups.

we need the following terminology about sheaves of simplicial homtopy groups:

  • for XX a simplicial set, write π 0(X)\pi_0(X) for its set of connected components and π n(X,x)\pi_n(X,x), n1n \geq 1, xX 0x \in X_0, for its nnth simplicial homotopy group at xx (the homotopy group of its geometric realization), π n(X,x)=π n(X,x)\pi_n(X,x) = \pi_n(|X|, x). This yields functors π 0:SimpSetSet\pi_0 : SimpSet \to Set and π n:SimpSetGrps\pi_n : SimpSet \to Grps.

  • By postcomposition these functors induce functors π 0:SimpSet S opSet S op \pi_0 : SimpSet^{S^{op}} \to Set^{S^{op}} and π n:SimpSet S opGrps S op \pi_n : SimpSet^{S^{op}} \to Grps^{S^{op}} .

  • By postcomposition with the sheafification functor this yields functors π˜ 0:SimpSet S opSh(S) \tilde \pi_0 : SimpSet^{S^{op}} \to Sh(S) and π˜ n:SimpSet S opSh(S,Grps) \tilde \pi_n : SimpSet^{S^{op}} \to Sh(S,Grps) .

Definition

A local weak equivalence of simplicial presheaves is a morphism f:XYf : X \to Y such that

  1. the morphism π˜ 0(f):π˜ 0Xπ˜ 0Y\tilde \pi_0(f) : \tilde \pi_0 X \to \tilde \pi_0 Y is an isomorphism in Sh(S)Sh(S);

  2. the diagrams

    π˜ nX π˜ nY X˜ 0 Y˜ 0 \array{ \tilde \pi_n X &\to& \tilde \pi_n Y \\ \downarrow && \downarrow \\ \tilde X_0 &\to& \tilde Y_0 }

    are pullback diagrams in Sh(S,SimpSet)Sh(S,SimpSet), for all n1n \geq 1, where X˜ 0\tilde X_0 denotes the sheaf associated to X 0X_0.

Equivalently a morphism f:XYf : X \to Y of simplicial presheaves is, equivalently, a local weak equivalence if all induced morphisms of sheaves

π˜ n(X U,x)π˜ nY U,f(x) \tilde \pi_n (X|_U, x) \to \tilde \pi_n Y|_U, f(x)

are isomorphisms for all USU \in S, for X U,Y UX|_U, Y|_U the pullbacks to the over-category site S/US/U, for all xX 0(U)x \in X_0(U) and all n0n \geq 0.

If the site SS has enough points then this condition is equivalent to saying that ff is a weak equivalence in the model structure on simplicial sets over every stalk (see p. 363 of Jardine01).

In terms of local liftings

(see DI02, i.e. Dugger and Isaksen, Weak equivalences of simplicial presheaves )

If XX and YY are local fibrations there is a characterisation in terms of local homotopy liftings. Write PP for the pushout of the diagram Δ nΔ n×Δ 1Δ n×Δ 1\partial \Delta^n \leftarrow \partial \Delta^n\times \Delta^1 \rightarrow \Delta^n\times \Delta^1. Then there are two maps Δ nP\Delta^n\rightarrow P by restriction of Δ n×Δ 1P\Delta^n\times \Delta^1\rightarrow P along the cofaces.

Then a local weak equivalence is a morphism f:XYf : X \to Y such that for all commuting diagrams

Δ nU X i U f Δ nU Y \array{ \partial \Delta^n \otimes U &\to& X \\ \downarrow^{i_U} && \downarrow^f \\ \Delta^n \otimes U &\to& Y }

with UU simplicially constantly representable there exists a covering sieve RR of UU such that for every VRV\in R there are morphisms g:Δ nVXg:\Delta^n \otimes V\rightarrow X and h:PVYh:P\otimes V\rightarrow Y for which gi V=Δ nVΔ nUXg\circ i_V=\partial\Delta^n \otimes V\rightarrow \partial\Delta^n \otimes U \rightarrow X and Δ nVΔ nUY=Δ nVPVY\Delta^n\otimes V \rightarrow \Delta^n \otimes U\rightarrow Y= \Delta^n\otimes V\rightarrow P\otimes V\rightarrow Y and in addition the square

Δ nV X f PU Y \array{ \Delta^n\otimes V &\to& X \\ \downarrow && \downarrow^f \\ P \otimes U &\to& Y }

commutes.

Examples

  • Every object-wise weak equivalence is in particular a local weak equivalence.

Local injective model structure

The local injective model structure on simplicial presheaves on a site CC is the left Bousfield localization SPr(C) locinjSPr(C)_{loc inj} of the injective global model structure SPr(C) injSPr(C)_{inj} at the class of local weak equivalences described above.

So

  • cofibrations are precisely the objectwise cofibrations of simplicial sets, i.e. the monomorphisms in SPr(S)SPr(S);

  • weak equivalences are the local weak equivalences from above.

Theorem

The inclusion of sheaves into simplicial presheaves SimpSh(S)SimpPr(S)SimpSh(S) \hookrightarrow SimpPr(S) and the sheafification functor SimpPr(S)SimpSh(S)SimpPr(S) \to SimpSh(S) constitute a Quillen equivalence with respect to the above local injective model structure on SimpPr(S)SimpPr(S) ans the local model structure on simplicial sheaves.

Proof

See Jardine07, theorem 5.

Theorem

The fibrant objects in the local injective model structure SPr(C) locinjSPr(C)_{loc inj} are those simplicial presheaves that

  1. are fibrant in the global injective model structure;

  2. satisfy descent for all hypercovers.

Proof

DHI04, theorem 1.1

Local projective model structure

The local projective model structure on simplicial presheaves on a site CC is the left Bousfield localization SPr(C) locprojSPr(C)_{loc proj} of the projective global model structure SPr(C) projSPr(C)_{proj} at the class of local weak equivalences described above.

So

  • cofibrations are precisely the cofibrations in the global projective structure (defined by left lifting property with respect to global Kan fibrations)

  • weak equivalences are the local weak equivalences from above.

Remark. Notice that this is still using left Bousfield localization. If we used right Bousfield localization the local projective fibrations would simply be the global Kan fibrations. Instead we have the following.

Theorem

The local injective model structure SPr(C) locinjSPr(C)_{loc inj} is Quillen equivalent to the “universal homtopy thepory” UC/SU C/S constructed by

  1. formally adding homotopy colimits to the category CC to create the category UCU C.

  2. imposing relations requiring that for every hypercover UXU \to X, the morphism hocolim nU nXhocolim_n U_n \to X is a weak equivalence.

Proof

DHI04, theorem 1.2

In UC/SU C/S the fibrant objects have a simpler description than in SPr(C) locinjSPr(C)_{loc inj}: they still need to satisfy descent but the implicit fibrancy condition with respect to the global injective structure is replaced by the fibrancy condition with respect to the global projective structure

Theorem

The fibrant objects in UC/SU C/S are those simplicial presheaves AA that

  1. are objectwise fibrant (i.e. take values in Kan complexes)

  2. satisfy descent for all hypercovers.

Proof

DHI04, theorem 1.3

Intermediate model structures

One can regard the projective and the injective model structure as two extrema of a poset of model structures on simplicial presheaves.

See Jardine07 page 12

References

See model structure on simplicial presheaves.

The local projective model structure on simplicial presheaves appears as theorem 1.6 in

  • Benjamin Blander, Local projective model structure on simplicial presheaves (pdf)

Its analog for sheaves, theorem 2.1 there, is due to

  • K. Brown, Gersten,

That the local projective model structure (directly defined) is indeed the left Bousfield localization of the global projective model structure is lemma 4.3 there.

Revised on August 17, 2011 07:16:33 by Beren Sanders (131.179.52.133)