nLab category of sheaves

Idea

This entry is about the properties and the characterization of the category $Sh(S)$ of (set-valued) sheaves on a (small) site $S$, which is a Grothendieck topos. Among other things it gives a definition and a characterization of the notion of sheaf itself, but for more details on sheaves themselves see there.

Definition

Let $(C,J)$ be a site: a (small) category equipped with a coverage.

Definition

The category of sheaves on $(C,J)$ is the full subcategory of the category of presheaves

$$i : Sh_J(C) \hookrightarrow PSh(C)$$

of those presheaves which are sheaves with respect to $J$.

Proposition

Every category of sheaves is a reflective subcategory

$$(L \dashv i) : Sh_J(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} Psh(C) \,,$$

hence a subtopos of the presheaf topos. Moreover, every such subtopos arises in this way: there is a bijection between Grothendieck topologies on $C$ and equivalence classes of geometric embeddings into $PSh(C)$.

This appears for instance as (Johnstone, corollary C.2.1.11). See also Lawvere-Tierney topology.

Proof

Details on the first statement are at sheafification. A full proof for the second statement is at (∞,1)-category of (∞,1)-sheaves (there proven in (∞,1)-category theory, but the proof is verbatim the same in category theory).

Equivalent characterizations

As localizations

Proposition

The category of sheaves is equivalent to the homotopy category of the category with weak equivalences $PSh(C)$ with the weak equivalences given by $W =$local isomorphisms

$$Sh(S) \simeq Ho_{PSh(S)} = PSh(C)[\text{local isomorphisms}]^{-1} \,.$$

The converse is also true: for every left exact functor $L : PSh(S) \to PSh(S)$ (preserving finite limits) which is left adjoint to the inclusion of its image, there is a Grothendieck topology on $S$ such that the image of $L$ is the category of sheaves on $S$ with respect to that topology.

We spell out proofs of some of the above claims.

Let $C$ be a small category and

$$i : \mathcal{E} \hookrightarrow [C^{op}, Set]$$

a reflective subcategory, hence a full subcategory with a left adjoint $L : [C^{op}, Set] \to \mathcal{E}$, such that moreover $L$ preserves finite limits.

Write $W := L^{-1}(isos) \subset Mor([C^{op}, Set])$ for the class of morphisms in $[C^{op}, Set]$ that are sent to isomorphisms by $L$.

Proposition

A presheaf $A \in [C^{op}, Set]$ is in $\mathcal{E}$ (meaning: in the essential image of $i$) precisely if for all $f : X \to Y$ in $W$ the induced function

$$Hom(f,A) : Hom(Y,A) \to Hom(X,A)$$

is a bijection.

Proof

If $A \simeq i \hat A$, then by the $(L \dashv i)$-adjunction isomorphism we have

$$Hom(f, i \hat A) \simeq \mathcal{E}(L(f), A) \,.$$

But by assumption $L(f)$ is an isomorphism, so the claim is immediate.

Conversely, if for all $f$ the function $Hom(f,A)$ is a bijection, define $\hat A := L(A)$ and let $\epsilon_A : A \to i L(A)$ be the $(L \dashv i)$-unit.

By the assumption that $i$ is a full and faithful functor and basic properties of adjoint functors we have that the counit

$$L i \to Id$$

is a natural isomorphism. By the zig-zag law the composite

$$L A \stackrel{L \epsilon_A}{\to} L i L A \stackrel{\simeq}{\to} L A$$

is the identity and therefore $L \epsilon$ is an isomorphism and so $\epsilon_A$ is in $W$, under our assumption on $A$.

Using this it follows that

$$Hom(\epsilon_A, A) : Hom(i L A, A) \stackrel{\simeq}{\to} Hom(A,A)$$

is an isomorphism. Write $k_A : i L A \to A$ for the preimage of $id_A$ under this isomorphism, which is therefore a left inverse of $\epsilon_A$. This immediately implies that also $k_A$ is in $W$, and so we can enter the same argument with $k_A$ to find that it has a left inverse itself. But this means that $k_A$ is in fact an isomorphism and hence so is $\epsilon A$, which thus exhibits $A$ as being in the essential image of $i$.

Proposition

A morphism $f : X \to Y$ is in $W$ precisely if for every morphism $z : j(c) \to Y$ with representable domain, the pullback $z^* f$ in

$$\array{ X \times_Y j(c) &\to& X \\ {}^{\mathllap{z^* f}}\downarrow && \downarrow^{\mathrlap{f}} \\ j(c) &\stackrel{z}{\to}& Y }$$

is in $W$.

Proof

Assume first that $f$ is in $W$. Since by assumption $L$ preserves finite limits, it follows that

$$\array{ L(X \times_Y j(c)) &\to& L X \\ {}^{\mathllap{L(z^* f)}}\downarrow && \downarrow^{\mathrlap{L f}} \\ L(j(c)) &\stackrel{L z}{\to}& L Y }$$

is still a pullback diagram in $\mathcal{E}$ and hence that $L(z^* f)$ is the pullback of the isomorphism $L f$ and thus itself an isomorphism. Therefore $z^* f$ is in $W$.

Conversely, suppose that all these pullbacks are in $W$. Then use the “co-Yoneda lemma” to write the presheaf $Y$ as a colimit over all representables mapping into it

$${\lim_\to}_{j(c_i) \stackrel{z_i}{\to} Y} j(c) \stackrel{\simeq}{\to} Y \,.$$

Forming the pullback along $f$, using that in a topos (such as our presheaf topos) colimits are preserved by pullbacks, we get

$$\array{ {\lim_\to}_i f^* j(c_i) &\stackrel{\simeq}{\to}& X \\ {}^{\mathllap{{\lim_\to}_i z_i^* f}}\downarrow && \downarrow^{\mathrlap{f}} \\ {\lim_\to}_i j(c_i) &\stackrel{\simeq}{\to}& Y } \,.$$

Since $L$ preserves all colimits and finite limits, we also get

$$\array{ {\lim_\to}_i L(f^* j(c_i)) &\stackrel{\simeq}{\to}& L(X) \\ {}^{\mathllap{{\lim_\to}_i L(z_i^* f)}}\downarrow && \downarrow^{\mathrlap{L(f)}} \\ {\lim_\to}_i L(j(c_i)) &\stackrel{\simeq}{\to}& L(Y) } \,.$$

Since by assumption now all $L(z_i^* f )$ are isomorphisms, also ${\lim_\to}_i L(z_i^* f)$ is an isomorphism and hence three sides of the above square are isomorphisms. Therefore also $L(f)$ is and hence $f$ is in $W$.

Proposition

The collection of sieves in $W$, hence the collection of monomorphisms in $W$ whose codomain is a representable, constitute a Grothendieck topology on $C$.

Proof

We check the list of axioms, given at Grothendieck topology:

1. Pullbacks of covering sieves are covering :

   First of all, the pullback of a sieve along a morphism of representables is still a sieve, because monomorphisms are (as discussed there) stable under pullback.

   Next, since $L$ preserves finite limits, $L$ applied to the pullback sieve is the pullback of an isomorphism, hence is an isomorphism, hence the pullback sieve is in $W$.

2. The maximal sieve is covering. Clear: $L$ applied to an isomorphism is an isomorphism.

3. Two sieves cover precisely if their intersection covers. This is again due to the pullback-stability of elements of $W$, due to the preservation of finite limits by $L$.

4. If all pullbacks of a sieve along morphisms of a covering sieve are covering, then the original sieve was covering .

   This is the same argument as in the second part of the proof of prop. .

Proposition

$\mathcal{E}$ is a Grothendieck topos.

Proof

By prop. and prop. we are reduced to showing that an object $A$ is in $\mathcal{E}$ already if for all monomorphisms $f$ in $W$ the function $Hom(f,A)$ is a bijection.

(…)

As toposes

Categories of sheaves are examples of categories that are toposes: they are the Grothendieck toposes characterized among all toposes as those satisfying Giraud's axioms.

Proposition

Sheaf toposes are equivalently the subtoposes of presheaf toposes.

This appears for instance as (Johnstone, corollary C.2.1.11).

As accessible reflective subcategories

Proposition

sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes

See also at reflective sub-(∞,1)-category.

Properties

Dependence on the site

Definition

For $(L \dashv i) : \mathcal{E} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}$ a category of sheaves and $(C,J)$ a site such that we have an equivalence of categories $\mathcal{E} \simeq Sh_J(C)$ we say that $C$ is a site of definition for the topos $\mathcal{E}$.

Remark

There are always different sites $(C,J)$ whose categories of sheaves are equivalent. First of all for fixed $C$ and given a coverage $J$, the category of sheaves depends only on the Grothendieck topology generated by $J$. But there may be site structures also on inequivalent categories $C$ that have equivalent categories of sheaves.

Definition

For $(C,J)$ a site with coverage $J$ and $D \to C$ any subcategory, the induced coverage $J_D$ on $D$ has as covering sieves the intersections of the covering sieves of $C$ with the morphisms in $D$.

Definition

Let $(C,J)$ be a site (possibly large). A subcategory $D \to C$ (not necessarily full) is called a dense sub-site with the induced coverage $J_D$ if

1. every object $U \in C$ has a covering $\{U_i \to U\}$ in $J$ with all $U_i$ in $D$;

2. for every morphism $f : U \to d$ in $C$ with $d \in D$ there is a covering family $\{f_i : U_i \to U\}$ such that the composites $f \circ f_i$ are in $D$.

Remark

If $D$ is a full subcategory then the second condition is automatic.

Theorem

(comparison lemma)

Let $(C,J)$ be a (possibly large) site with $C$ a locally small category and let $f : D \to C$ be a small dense sub-site. The pair of adjoint functors

$$(f^* \dashv f_*) : PSh(D) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} PSh(C)$$

with $f^*$ given by precomposition with $f$ and $f_*$ given by right Kan extension induces an equivalence of categories between the categories of sheaves

$$(f_* \dashv f^*) : Sh_{J_D}(D) \underoverset {\underset{f_*}{\to}}{\overset{f^*} {\leftarrow}} {\simeq} Sh_J{C} \,.$$

This appears as (Johnstone, theorm C2.2.3).

Examples

• Let $X$ be a locale with frame $Op(X)$ regarded as a site with the canonical coverage ($\{U_i \to U\}$ covers if the join of the $U_i$ is $U$). Let $bOp(X)$ be a basis for the topology of $X$: a complete join-semilattice such that every object of $Op(X)$ is the join of objects of $bOp(X)$. Then $bOp(X)$ is a dense sub-site.

  • For $X$ a locally contractible space, $Op(X)$ its category of open subsets and $cOp(X)$ the full subcategory of contractible open subsets, we have that $cOp(X)$ is a dense sub-site.

• For $C = TopManifold$ the category of all paracompact topological manifolds equipped with the open cover coverage, the category CartSp${}_{top}$ is a dense sub-site: every paracompact manifold has a good open cover by open balls homeomorphic to a Cartesian space.

• Similaryl for $C =$ Diff the category of smooth manifolds equipped with the good open cover coverage, the full subcategory CartSp${}_{smooth}$ is a dense sub-site.

Limits and colimits

Beware that, in general, constant presheaves need not be sheaves. But the presheaf constant on the singleton set is always a sheaf:

Example

(terminal sheaf)
For every site $\mathcal{S}$, the terminal object in its category of sheaves $Sh(\mathcal{S})$ is, as a presheaf, the constant functor with value the singleton set.

Proof

Recalling the defining geometric embedding of any Grothendieck topos into the corresponding category of presheaves

$$Sh(\mathcal{S}) \underoverset {\underset{i}{\hookrightarrow}} {\overset{L}{\longleftarrow}} {\;\;\; \bot \;\;\;} PSh(\mathcal{S}) \,,$$

both functors preserve the terminal object (this being the limit over the empty diagram): The direct image because it is a right adjoint and right adjoints preserve limits, and the inverse image because sheafification preserves finite limits.

But the terminal presheaf is clearly the functor $\const_\ast \,\in\, PSh(\mathcal{S})$ which is constant on the singleton set. Therefore $L(const_\ast)$ is the terminal sheaf. But then its underlying presheaf $i\big(L(const_\ast)\big)$ must again be the terminal presheaf, hence still the functor constant on the singleton set.

Epi- mono- and isomorphisms

Proposition

Let $\mathcal{C}$ be a small site and let $Sh(\mathcal{C})$ be its category of sheaves. Let $f \colon X \to Y$ be a homomorphism of sheaves, hence a morphism in $Sh(\mathcal{C})$. Then:

1. $f$ is a monomorphism or isomorphism precisely if it is so globally in that for each object $U \in \mathcal{C}$ in the site, then the component $f_U \colon X(U) \to Y(U)$ is an injection or bijection of sets, respectively.

2. $f$ is an epimorphism precisely if it is so locally, in that: for all $U \in C$ and every element $y \in Y(U)$ there is a covering $\{p_i : U_i \to U\}_{i \in I}$ such that for all $i \in I$ the element $Y(p_i)(y)$ is in the image of $f(U_i) : X(U_i) \to Y(U_i)$.

But if $\{x_i\}_{i \in I}$ is a set of points of a topos for $Sh(\mathcal{C})$ such that these are enough points (def.) then the morphism $f$ is epi/mono/iso precisely it is is so an all stalks, hence precisely if

$$x_i^\ast f \;\colon\; x_i^\ast X \longrightarrow x_i^\ast Y$$

is a surjection/injection/bijection of sets, respectively, for all $i \in I$.

Exactness properties

Every sheaf topos satisfies the following exactness properties. it is an

• extensive category;

• adhesive category;

• exhaustive category.

Related concepts

• coverage

  trivial coverage

• category of sheaves

• category of sheaves on a topological space

• (∞,1)-category of (∞,1)-sheaves.

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Gabriel–Ulmer’s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

References

• Francis Borceux, vol 3 of Handbook of Categorical Algebra, Cambridge University Press (1994)

• Peter Johnstone, sections A.4 and C.2 in Sketches of an Elephant

• The Stacks Project, Sites and sheaves (pdf)

The characterization of sheaf toposes and Grothendieck topologies in terms of left exact reflective subcategories of a presheaf category is also in

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

where it is implied by the combination of Corollary VII 4.7 and theorem V.4.1.

There is also a direct proof in Shane Kelly, What is the relationship between Grothendieck pretopologies and Grothendieck topologies? (web) .

The characterization of $Sh(S)$ as the homotopy category of $PSh(S)$ with respect to local isomorphisms is emphasized at the beginning of the text

• Bertrand Toen, Stacks and non-abelian cohomology (web) .

Details are in

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves (2006)

It’s a bit odd that the full final statement does not seem to be stated there explicitly, but all the ingredients are discussed:

• exercise 16.7 shows that sheafification inverts precisely the local isomorphisms, so that in particular every local isomorphism between sheaves is an isomorphism;

• lemma 16.3.2 states that the unit of the adjunction $Id_{PSh(S)} \rightarrow i \circ \bar{(-)} : PSh(S) \to PSh(S)$ is componentwise a local isomorphism;

• using this corollary 7.2.2 says that $Sh(S) \simeq Ho_{PSh(S)}$ with the homotopy category $Ho_{PSh(S)}$ formed using local isomorphisms as weak equivalences.

Letcure notes:

• Pierre Schapira, An Introduction to Categories and Sheaves, lecture notes (2023) [pdf, pdf]