nLab
(infinity,1)-category of (infinity,1)-sheaves

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Locality and descent

Contents

Idea

The notion of (,1)(\infty,1)-category of (,1)(\infty,1)-sheaves is the generalization of the notion of category of sheaves from category theory to the higher category theory of (∞,1)-categories.

Definition

Definition

An (,1)(\infty,1)-category of (,1)(\infty,1)-sheaves is a reflective sub-(∞,1)-category

Sh(C)LPSh(C) Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)

of an (∞,1)-category of (∞,1)-presheaves such that the following equivalent conditions hold

This is HTT, def. 6.2.2.6.

An (,1)(\infty,1)-category of (,1)(\infty,1)-sheaves is an (∞,1)-topos.

Remark

Equivalence (1) is the descent condition and the presheaves satisfying it are the (∞,1)-sheaves .

Typically UU here is the Cech nerve

C({U i})=lim [n]U i 0,U i n C(\{U_i\}) = \lim_{\to_{[n]}} U_{i_0, \cdots U_{i_n}}

of a covering family {U ic}\{U_i \to c\} (where the colimit is the (∞,1)-categorical colimit or homotopy colimit) so that the above descent condition becomes

A(c)PSh(lim U ,A)lim A(U )=lim ( i,jA(U i)× A(c)A(U j) iA(U i)). A(c) \simeq PSh(\lim_\to U_\cdots, A) \simeq \lim_{\leftarrow} A(U_\cdots) = \lim_{\leftarrow} \left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} \prod_{i,j} A(U_i) \times_{A(c)} A(U_j) \stackrel{\to}{\to}\prod_i A(U_i) \right) \,.
Remark

Sometimes (∞,1)-sheaves are called ∞-stacks, though sometimes the latter term is reserved for hypercomplete (,1)(\infty,1)-sheaves and at other times again it refers to (∞,2)-sheaves.

The (n,1)-categorical counting is:

  • sheaf = 0-stack = 0-truncated (,1)(\infty,1)-sheaf

  • (2,1)(2,1)-sheaf = stack = 1-truncated (,1)(\infty,1)-sheaf

  • (3,1)(3,1)-sheaf = 2-stack = 2-truncated (,1)(\infty,1)-sheaf

  • etc.

  • (,1)(\infty,1)-sheaf = ∞-stack (or = hypercomplete (,1)(\infty,1)-sheaf).

Properties

Localizations and Grothendieck topology

We reproduce the proof that the two characterization in def. 1 above are indeed equivalent.

Proposition

For CC an (∞,1)-site, the full sub-(∞,1)-category of PSh(C)PSh(C) on local objects with respect to the covering monomorphisms in PSh(C)PSh(C) is indeed a topological localization, and hence Sh(C)Sh(C) is indeed an exact reflective sub-(∞,1)-category of PSh(C)PSh(C) and hence an (∞,1)-topos.

This is HTT, Prop. 6.2.2.7

Proof

We must prove that the (∞,1)-sheafification functor L:PSh(C)Sh(C)L \colon PSh(C)\to Sh(C) preserves finite (∞,1)-limits. To do so we give an explicit construction of LL. Given a presheaf FPSh(C)F\in PSh(C), define a new presheaf F +F^+ by the formula

F +(c)=lim Ulim uUF(u)F^+(c)={\lim_{\rightarrow}}_U {\lim_{\leftarrow}}_{u\in U} F(u)

where the colimit is taken over all covering sieves UU of cc; this is called the plus construction. It defines a functor PSh(C)PSh(C)PSh(C)\to PSh(C) and there is an obvious morphism FF +F\to F^+ natural in FF.

It is clear that the construction FF +F\mapsto F^+ preserves finite (∞,1)-limits, since filtered (∞,1)-colimits do, and it is easy to see that the map FF +F\to F^+ becomes an equivalence in Sh(C)Sh(C). Given an ordinal α\alpha, let F (α)F^{(\alpha)} be the α\alpha-iteration of the plus construction applied to the presheaf FF. Then the functor FF (α)F\mapsto F^{(\alpha)} preserves finite limits and the canonical map FF (α)F\to F^{(\alpha)} becomes an equivalence in Sh(C)Sh(C). In particular, if F (α)F^{(\alpha)} is a sheaf, then F (α)L(F)F^{(\alpha)}\simeq L(F). Thus, it suffices to show that there exists an ordinal α\alpha such that, for every FPSh(C)F\in PSh(C), F (α)F^{(\alpha)} is a sheaf.

Fix cCc\in C and a covering sieve UU of CC. Given a presheaf GPSh(C/c)G\in PSh(C/c), we define an auxiliary presheaf Match(U,G)PSh(C/c)Match(U,G)\in PSh(C/c) by the formula

Match(U,G)(f:dc)=lim uf *UG(u).Match(U,G)(f: d\to c)={\lim_{\leftarrow}}_{u\in f^\ast U}G(u).

Restriction maps induce a morphism θ G:GMatch(U,G)\theta_G: G\to Match(U,G). Since we clearly have G(u)Match(U,G)(u)G(u)\stackrel{\sim}{\to} Match(U,G)(u) for uUu\in U, the functor Match(U,)Match(U,-) is idempotent in the sense that Match(U,θ G)Match(U,\theta_G) and θ Match(U,G)\theta_{Match(U,G)} are (equivalent) equivalences.

By definition, FPSh(C)F\in PSh(C) is a sheaf if and only if F(c)Match(U,F| C/c)(c)F(c)\stackrel{\sim}{\to} Match(U,F|_{C/c})(c) for every cCc\in C and every covering sieve UU of cc. Our goal is therefore to find an ordinal α\alpha (depending only on the (∞,1)-site CC) such that, for every FPSh(C)F\in PSh(C), the map

F (α)(c)Match(U,F (α)| C/c)(c)F^{(\alpha)}(c) \to \Match(U,F^{(\alpha)}|_{C/c})(c)

is an equivalence.

The morphism GG +G\to G^+ in PSh(C/c)PSh(C/c) factors as

GMatch(U,G)G +.G\to Match(U,G)\to G^+.

Applying Match(U,)Match(U,-) to this factorization, we get a commutative diagram

G Match(U,G) G + θ G θ Match(U,G) θ G + Match(U,G) Match(U,Match(U,G)) Match(U,G +) \array{ G &\to& Match(U,G) &\to& G^+ \\ \downarrow^{\mathrlap{\theta_G}} && \downarrow^{\mathrlap{\theta_{Match(U,G)}}} && \downarrow^{\mathrlap{\theta_{G^+}}} \\ Match(U,G) &\to& Match(U,Match(U,G)) &\to& Match(U,G^+) }

in which the map θ Match(U,G)\theta_{Match(U,G)} is an equivalence since Match(U,)Match(U,-) is idempotent. By cofinality, the colimit of the maps θ G (n)\theta_{G^{(n)}} as nn\to\infty is an equivalence. Applying this to G=F| C/cG=F|_{C/c}, we get

F (ω)(c)lim nMatch(U,F (n)| C/c)(c). F^{(\omega)}(c)\stackrel{\sim}{\to} {\lim_{\rightarrow}}_{n\to\infty} Match(U,F^{(n)}|_{C/c})(c).

This almost means that F (ω)F^{(\omega)} is a sheaf. The problem is that the filtered colimit on the right-hand side need not commute with the limit appearing in the definition of Match(U,)Match(U,-), that is, the canonical map

lim α<ωMatch(U,F (α)| C/c)(c)Match(U,F (ω)| C/c)(c) {\lim_{\rightarrow}}_{\alpha \lt \omega} Match(U,F^{(\alpha)}|_{C/c})(c) \to \Match(U,F^{(\omega)}|_{C/c})(c)

need not be an equivalence. To solve this problem, we choose a cardinal κ\kappa such that for every cCc\in C and every covering sieve UU of cc, the functor Match(U,()| C/c)(c):Psh(C)GrpdMatch(U,(-)|_{C/c})(c):Psh(C)\to \infty Grpd preserves κ\kappa-filtered colimits. This is possible because CC is small and each of these functors, being the composition of the restriction functor PSh(C)PSh(U)PSh(C)\to PSh(U) and the limit functor PSh(U)GrpdPSh(U)\to\infty Grpd, has a left adjoint (∞,1)-functor and is therefore accessible (see HTT Prop. 5.4.7.7). Then the above map with ω\omega replaced by κ\kappa is an equivalence. For every ordinal α<κ\alpha\lt\kappa, applying the above to F (α)F^{(\alpha)} shows that

F (α+ω)(c)lim nMatch(U,F (α+n)| C/c)(c), F^{(\alpha+\omega)}(c)\stackrel{\sim}{\to} {\lim_{\rightarrow}}_{n\to\infty} Match(U,F^{(\alpha+n)}|_{C/c})(c),

Since κ\kappa is a limit ordinal, we deduce that F (κ)F^{(\kappa)} is a sheaf by cofinality.

And conversely:

Proposition

(equivalence of site structures and categories of sheaves)

For CC a small (∞,1)-category, there is a bijective correspondence between structure of an (∞,1)-site on CC and equivalence classes of topological localizations of PSh(C)PSh(C).

This is HTT, prop. 6.2.2.9.

Lemma

For CC a small (∞,1)-site and Sh(C)LPSh(C)Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C) the corressponding reflective inclusion of (∞,1)-sheaves into (∞,1)-presheaves on CC we have that the image under LL of a sub-(,1)(\infty,1)-functor p:Uj(c)p : U \to j(c) of a representable j(c)j(c) is covering precisely if L(p)L(p) is an equivalence.

This is HTT, lemma 6.2.2.8.

Proof of the Lemma

Since Sh(C)Sh(C) is the reflectuive localization of PSh(C)PSh(C) at covering monomorphisms, it is clear that if p:Uj(c)p : U \to j(c) is covering, then L(p)L(p) is an equivalence.

To see the converse, form the 0-truncation of LiL i and conclude as for ordinary sheaves on the homotopy catgegory of CC.

Proof of the Proposition

We have seen in (…) that for every structure of an (,1)(\infty,1)-site on CC we obtain a topological localization of the presheaf category, and that this is an injective map from site structures to equivalence classes of sheaf categories. It remains to show that it is also a surjective map, i.e. that every topological localization of PSh(C)PSh(C) comes from the structure of an (∞,1)-site on CC.

So consider SMor(PSh(C))S \subset Mor(PSh(C)) a strongly saturated class of morphisms which s topological (closed under pullbacks). Write S 0SS_0 \subset S for the subcalss of those that are monomorphisms of the form Uj(c)U \to j(c).

Observe that then SS is indeed generated by (is the smallest strongly saturated class containing) S 0S_0: since by the co-Yoneda lemma every object XPSh(C)X \in PSh(C) is a colimit xlim kj(Ξ k)x \simeq {\lim_\to}_k j(\Xi_k) over representables. It follows that every monomorphism f:YXf : Y \to X is a colimit (in Func(Δ[1],PSh(C))Func(\Delta[1],PSh(C))) of those of the form Uj(c)U \to j(c): for consider the pullback diagram

f *(lim kΞ k) Y f f lim kΞ k X(lim kf *Ξ k) Y f f lim kΞ k X \array{ f^* ({\lim_\to}_k \Xi_k) &\to& Y \\ \downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}} \\ {\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X } \;\;\;\;\; \simeq \;\;\;\;\; \array{ ({\lim_\to}_k f^* \Xi_k) &\to& Y \\ \downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}} \\ {\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X }

where the equivalence is due to the fact that we have universal colimits in PSh(C)PSh(C). This realizes ff as a colimit over morphisms of the form f *j(Ξ k)j(Ξ k)f^* j(\Xi_k) \to j(\Xi_k) that are each a pullback of a monomorphism. Since monomorphisms are stable under pullback (see monomorphism in an (∞,1)-category for details), all these component morphisms are themselves monomorphisms.

So every monomorphism in SS is generated from S 0S_0, but by the assumption that SS is topological, it is itself entirely generated from monomorphisms, hence is generated from S 0S_0.

So far this establishes that evry topological localization of PSh(C)PSh(C) is a localization at a collection of sieves/ subfunctors Uj(c)U \to j(c) of representables. It remains to show that this collection of subfunctors is indeed an Grothendieck topology and hence exhibits on CC the structure of an (∞,1)-site. We check the necessary three axioms:

  1. equivalences cover – The equivalences j(c)j(c)j(c) \stackrel{\simeq}{\to} j(c) belong to SS and are monomorphisms, hence belong to S 0S_0.

  2. pullback of a cover is covering - Since monomorphisms are stable under pullback, we haave for every p:Uj(c)p : U \to j(c) in SS and every j(f):j(d)j(c)j(f) : j(d) \to j(c) that also the pullback f *pf^* p

    f *U U f *p p j(d) f j(c) \array{ f^* U &\to& U \\ \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ j(d) &\stackrel{f}{\to}& j(c) }

    is a monomorphism and in SS, hence in S 0S_0.

  3. if restriction of a sieve to a cover is covering, then the sieve is covering – Let p:Uj(c)p : U \to j(c) be an arbitrary monomorphism and f:Xj(d)f : X \to j(d) in S 0S_0. Write Xlim kΞ kX \simeq {\lim_\to}_k \Xi_k and consider the pullback

    lim kp *Ξ k p *f U lim kf k *p p lim kΞ k f j(c), \array{ {\lim_\to}_k p^* \Xi_k &\stackrel{p^* f}{\to}& U \\ \downarrow^{{\lim_\to}_k f_k^* p} && \downarrow^{\mathrlap{p}} \\ {\lim_\to}_k \Xi_k &\stackrel{f}{\to}& j(c) } \,,

    where again we made use of the universal colimits in PSh(C)PSh(C). Now notice that

    1. ff is in SS by assumption;

    2. p *fp^* f is by pullback stability of SS;

    3. each of the f kpf_k p is in SS by assumption, hence lim kf k *p{\lim_k f_k^* p} is by the fact that SS is strongly saturated.

    4. so by commutativity pp *fp \circ p^*f is in SS.

    5. finally by 2-out-of-3 this means that pp is in SS.

Over paracompact topological spaces

We discuss how (,1)(\infty,1)-sheaves over a paracompact topological space are equivalent to topological spaces over XX. This the analogue of the 1-categorical statement that sheaves on XX are equivalent to etale spaces over XX: an etale space over XX is one whose fibers are discrete topological space, hence 0-truncated spaces. Then n-category analogy has homotopy n-types as fibers.

Definition

For YXY \to X a morphism in Top, and UOp(X)U \in Op(X) an open subset of XX, write

Sing X(Y,U):=Hom X(U×Δ ,X) Sing_X(Y,U) := Hom_X(U \times \Delta^\bullet, X)

for the simplicial set (in fact a Kan complex) of continuous maps

U×Δ k Y X \array{ U \times \Delta^k && \to && Y \\ & \searrow && \swarrow \\ && X }

form UU times the topological kk-simplex Δ k\Delta^k into YY, that are sections of YXY \to X.

This is a relative version of the singular simplicial complex functor.

Proposition

Let (X,)(X, \mathcal{B}) be a topological space equipped with a base for the topology \mathcal{B}.

There is a model category structure on the over category Top/XTop/X with weak equivalences and fibration precisely those morphisms YZY \to Z over XX such that for each UU \in \mathcal{B} the induced morphism Sing X(Y,U)Sing X(Z,U)Sing_X(Y,U) \to Sing_X(Z,U) is a weak equivalence or fibration, respectively, in the standar model structure on simplicial sets.

This is HTT, prop 7.1.2.1.

Write (Top/X) (Top/X)^\circ for the (∞,1)-category presented by this model structure.

Proposition

Let XX be a paracompact topological space and write as usual Sh (,1)(X):=Sh (,1)(Op(X))Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X)) for the (,1)(\infty,1)-category of (,1)(\infty,1)-sheaves on the category of open subsets of XX; equipped with the canonical structure of a site.

Let \mathcal{B} be the set of F σF_\sigma-open subsets of XX. This are those open subsets that are countable unions of closed subsets, equivalently the 0-sets of continuous functions X[0,1]X \to [0,1].

Let Top/X Top/X^\circ be the corresponding (,1)(\infty,1)-categoty according to the above proposition. Then Sing X(,)Sing_X(-,-) constitutes an equivalence of (∞,1)-categories

Top/X Sh (,1)(X). Top/X^\circ \simeq Sh_{(\infty,1)}(X) \,.

This is HTT, corollary 7.1.4.4.

Difference to more general (,1)(\infty,1)-toposes

The (∞,1)-toposes that are (,1)(\infty,1)-categories of sheaves, i.e. that arise by topological localization from an (∞,1)-category of (∞,1)-presheaves, enjoy a number of special properties over other classes of (,1)(\infty,1)-toposes, such as notably hypercomplete (∞,1)-toposes.

The following lists these properties. (HTT, section 6.5.4.)

Universal property

The construction of (∞,1)-sheaf (∞,1)-toposes on a given locale is singled out over the construction of other kinds of (,1)(\infty,1)-toposes (such as hypercomplete (∞,1)-toposes) by the following universal property:

forming (,1)(\infty,1)-sheaves is, roughly, right adjoint to the functor τ 1\tau_{\leq -1} that sends each (,1)(\infty,1)-topos to its underlying locale of subobjects of the terminal object.

See HTT, item 1) of section 6.5.4.

For X,YX,Y two (,1)(\infty,1)-toposes, write Geom(X,Y)Func(X,Y)Geom(X,Y) \subset Func(X,Y) for the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that are geometric morphisms.

Lemma

For CC an small (n,1)-category with finite (∞,1)-limits and equipped with the structure of an (∞,1)-site and for YY an (∞,1)-topos, the truncation functor

τ n1:Geom(Y,Sh(C))Geom(τ n1Y,τ n1Sh(C)) \tau_{\leq n-1} : Geom(Y, Sh(C)) \to Geom(\tau_{\leq n-1} Y, \tau_{\leq n-1} Sh(C))

is an equivalence (of (∞,1)-categories).

This is HTT, lemma 6.4.5.6.

See also n-localic (∞,1)-topos.

Compact generation

Proposition

Let XX be a coherent topological space and let Op(X)Op(X) be its category of open subsets with the standard structure of an (∞,1)-site.

Then Sh (,1)(X):=Sh (,1)(Op(X))Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X)) is compactly generated in that it is generated by filtered colimits of compact objects.

Moreover, the compact objects of Sh (,1)(X)Sh_{(\infty,1)}(X) are those that are stalkwise compact objects in ∞Grpd and locally constant along a suitable stratification of XX.

This is HTT, prop. 6.5.4.4.

This statement is false for the hypercompletion of Sh (,1)(X)Sh_{(\infty,1)}(X), in general.

Nonabelian cohomology

For XX a topological space, let

(LConstΓ):Sh (,1)(X)ΓLConst (LConst \dashv \Gamma) : Sh_{(\infty,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}

be the global sections terminal geometric morphism.

For AGrpdA \in \infty Grpd, the (nonabelian) cohomology of XX with coefficients in AA is usually defined in ∞Grpd as

H(X,A):=π 0Func(SingX,A), H(X,A) := \pi_0 Func(Sing X, A) \,,

where SingXSing X is the fundamental ∞-groupoid of XX. On the other hand, if we send AA into Sh (,1)(X)Sh_{(\infty,1)}(X) via LConstLConst, the there is the intrinsic cohomology of the (,1)(\infty,1)-topos Sh (,1)(X)Sh_{(\infty,1)}(X)

H(X,A):=π 0Sh (,1)(X)(X,LConstA). H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A) \,.

Noticing that XX is in fact the terminal object of Sh (,1)(X)Sh_{(\infty,1)}(X) and that Sh (,1)(X)(X,)Sh_{(\infty,1)}(X)(X,-) is in fact that global sections functor, this is equivalently

π 0ΓLConstA. \cdots \simeq \pi_0 \Gamma LConst A \,.
Theorem

If XX is a paracompact space, then these two definitins of nonabelian cohomology of XX with constant coefficients AGrpdA \in \infty Grpd agree:

H(X,A):=π 0Grpd(SingX,A)Sh (,1)(X)(X,LConstA). H(X,A) := \pi_0 \infty Grpd(Sing X,A) \simeq Sh_{(\infty,1)}(X)(X,LConst A) \,.

This is HTT, theorem 7.1.0.1.

Models

The topological localizations of an (∞,1)-category of (∞,1)-presheaves are presented by the left Bousfield localization of the global model structure on simplicial presheaves at the set of Cech covers.

The hypercomplete (,1)(\infty,1)-sheaf toposes are presented by the local Joyal-Jardine model structure on simplicial presheaves.

Detailed discussion of this model category presentation is at

References

The study of simplicial presheaves apparently goes back to

which considers locally Kan simplicial presheaves as a category of fibrant objects.

This was later conceived in terms of a model structure on simplicial presheaves and on simplicial sheaves by Joyal and Jardine. Toë summarizes the situation and emphasizes the interpretation in terms of ∞-stacks living in (,1)(\infty,1)-categories for instance in

B. Toën, Higher and derived stacks: a global overview (arXiv) .

This concerns mostly hypercomplete (,1)(\infty,1)-sheaves, though.

The full picture in terms of Grothendieck-(∞,1)-toposes of (∞,1)-sheaves is the topic of

  • Jacob Lurie, Higher Topos Theory .

    • localization (,1)(\infty,1)-functors ((,1)(\infty,1)-sheafification for the present purpose) are discussed in section 5.2.7;

    • local objects ((,1)(\infty,1)-sheaves for the present purpose) and local isomorphisms are discussed in section 5.5.4;

    • the definition of (,1)(\infty,1)-topoi of (,1)(\infty,1)-sheaves is then definition 6.1.0.4 in section 6.1;

    • the characterization of (,1)(\infty,1)-sheaves in terms of descent is in section 6.1.3

    • the relation between the Brown–Joyal–Jardine model and the general story is discussed at length in section 6.5.4

Revised on September 1, 2014 11:03:16 by Urs Schreiber (185.37.147.14)