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

Idea

The notion of $(\infty,1)$-category of $(\infty,1)$-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 $(\infty,1)$-category of $(\infty,1)$-sheaves is a reflective sub-(∞,1)-category

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

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

• $L$ is a topological localization;

• there is the structure of an (∞,1)-site on $C$ such that the objects of $Sh(C)$ are precisely those (∞,1)-presheaves $A$ that are local objects with respect to the covering monomorphisms $p : U \to j(c)$ in $PSh(C)$ in that

  (1)$$A(c) \simeq PSh(j(c),A) \stackrel{PSh(p,A)}{\to} PSh(U,A)$$

  is an (∞,1)-equivalence in ∞Grpd.

This is HTT, def. 6.2.2.6.

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

Remark

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

Typically $U$ here is the Cech nerve

$$C(\{U_i\}) = \lim_{\to_{[n]}} U_{i_0, \cdots U_{i_n}}$$

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

$$A(c) \simeq PSh(\lim_\to U_\cdots, A) \simeq \lim_{\leftarrow} A(U_\cdots) = \lim_{\leftarrow} \left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} \prod_{i,j} A(U_i) \times_{A(c)} A(U_j) \stackrel{\to}{\to}\prod_i A(U_i) \right) \,.$$

Remark

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

The (n,1)-categorical counting is:

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

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

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

• etc.

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

Properties

Localizations and Grothendieck topology

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

Proposition

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

This is HTT, Prop. 6.2.2.7

Proof

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

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

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

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

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

$$Match(U,G)(f: d\to c)={\lim_{\leftarrow}}_{u\in f^\ast U}G(u).$$

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

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

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

is an equivalence.

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

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

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

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

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

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

This almost means that $F^{(\omega)}$ 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 $Match(U,-)$, that is, the canonical map

$${\lim_{\rightarrow}}_{\alpha \lt \omega} Match(U,F^{(\alpha)}|_{C/c})(c) \to \Match(U,F^{(\omega)}|_{C/c})(c)$$

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

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

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

And conversely:

Proposition

(equivalence of site structures and categories of sheaves)

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

This is HTT, prop. 6.2.2.9.

Lemma

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

This is HTT, lemma 6.2.2.8.

Proof of the Lemma

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

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

…

Proof of the Proposition

We have seen in (…) that for every structure of an $(\infty,1)$-site on $C$ 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 $PSh(C)$ comes from the structure of an (∞,1)-site on $C$.

So consider $S \subset Mor(PSh(C))$ a strongly saturated class of morphisms which s topological (closed under pullbacks). Write $S_0 \subset S$ for the subcalss of those that are monomorphisms of the form $U \to j(c)$.

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

$$\array{ f^* ({\lim_\to}_k \Xi_k) &\to& Y \\ \downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}} \\ {\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X } \;\;\;\;\; \simeq \;\;\;\;\; \array{ ({\lim_\to}_k f^* \Xi_k) &\to& Y \\ \downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}} \\ {\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X }$$

where the equivalence is due to the fact that we have universal colimits in $PSh(C)$. This realizes $f$ as a colimit over morphisms of the form $f^* j(\Xi_k) \to j(\Xi_k)$ 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 $S$ is generated from $S_0$, but by the assumption that $S$ is topological, it is itself entirely generated from monomorphisms, hence is generated from $S_0$.

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

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

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

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

   is a monomorphism and in $S$, hence in $S_0$.

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

   $$\array{ {\lim_\to}_k p^* \Xi_k &\stackrel{p^* f}{\to}& U \\ \downarrow^{{\lim_\to}_k f_k^* p} && \downarrow^{\mathrlap{p}} \\ {\lim_\to}_k \Xi_k &\stackrel{f}{\to}& j(c) } \,,$$

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

   1. $f$ is in $S$ by assumption;

   2. $p^* f$ is by pullback stability of $S$;

   3. each of the $f_k p$ is in $S$ by assumption, hence ${\lim_k f_k^* p}$ is by the fact that $S$ is strongly saturated.

   4. so by commutativity $p \circ p^*f$ is in $S$.

   5. finally by 2-out-of-3 this means that $p$ is in $S$.

Over paracompact topological spaces

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

Definition

For $Y \to X$ a morphism in Top, and $U \in Op(X)$ an open subset of $X$, write

$$Sing_X(Y,U) := Hom_X(U \times \Delta^\bullet, X)$$

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

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

from $U$ times the topological $k$-simplex $\Delta^k$ into $Y$, that are sections of $Y \to X$.

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

Proposition

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

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

This is HTT, prop 7.1.2.1.

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

Proposition

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

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

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

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

This is HTT, corollary 7.1.4.4.

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

The (∞,1)-toposes that are $(\infty,1)$-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 $(\infty,1)$-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 $(\infty,1)$-toposes (such as hypercomplete (∞,1)-toposes) by the following universal property:

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

See HTT, item 1) of section 6.5.4.

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

Lemma

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

$$\tau_{\leq n-1} : Geom(Y, Sh(C)) \to Geom(\tau_{\leq n-1} Y, \tau_{\leq n-1} Sh(C))$$

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

This is HTT, lemma 6.4.5.6.

See also n-localic (∞,1)-topos.

Compact generation

Proposition

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

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

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

This is HTT, prop. 6.5.4.4.

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

Nonabelian cohomology

For $X$ a topological space, let

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

be the global sections terminal geometric morphism.

For $A \in \infty Grpd$, the (nonabelian) cohomology of $X$ with coefficients in $A$ is usually defined in ∞Grpd as

$$H(X,A) := \pi_0 Func(Sing X, A) \,,$$

where $Sing X$ is the fundamental ∞-groupoid of $X$. On the other hand, if we send $A$ into $Sh_{(\infty,1)}(X)$ via $LConst$, the there is the intrinsic cohomology of the $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$

$$H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A) \,.$$

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

$$\cdots \simeq \pi_0 \Gamma LConst A \,.$$

Theorem

If $X$ is a paracompact space, then these two definitions of nonabelian cohomology of $X$ with constant coefficients $A \in \infty Grpd$ agree:

$$H(X,A) := \pi_0 \infty Grpd(Sing X,A) \simeq Sh_{(\infty,1)}(X)(X,LConst A) \,.$$

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

Detailed discussion of this model category presentation is at

• models for (infinity,1)-category of (infinity,1)-sheaves .

Related concepts

• geometric homotopy type theory

References

The study of simplicial presheaves apparently goes back to

• K. Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology

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. Toën summarizes the situation and emphasizes the interpretation in terms of ∞-stacks living in $(\infty,1)$-categories for instance in

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

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

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

• Jacob Lurie, Higher Topos Theory .

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

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

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

  • the characterization of $(\infty,1)$-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

An overview is in

• Denis-Charles Cisinski, Catégories supérieures et théorie des topos, Séminaire Bourbaki, 21.3.2015, pdf.