# nLab category of sheaves

### Context

#### Topos Theory

Could not include topos theory - contents

# Contents

## 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

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

on 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 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

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

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 habe 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. 4.

###### Proposition

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

###### Proof

By prop. 5 and prop. 3 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 2.1.11).

### As accessible reflective subcategories

###### Proposition

Sheaf toposes are equivalently the accessible reflective subcategories of categories of presheaves.

## 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. Then 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 $U_i$ us $U$). 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.

### Epi- mono- and isomorphisms

Let $Sh(C)$ be a category of sheaves.

###### Proposition

A morphism $f : X \to Y$ is a monomorphism if it is a monomorphism of presheaves, that is if for each $U \in C$ the function $f(U) : X(U) \to Y(U)$ is injective.

###### Proposition

A morphism $f : X \to Y$ in $Sh(C)$ is an epimorphism precisely if it is locally surjective in the sense that:

for all $U \in C$ there is a covering $\{p_i : U_i \to U\}_{i \in I}$ such that for all $i \in I$ and every element $y \in Y(U)$ the element $f(p_i)(y)$ is in the image of $f(U_i) : X(U_i) \to Y(U_i)$.

### Exactness properties

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

• category of sheaves

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categoriessatisfying Giraud's axiomsinclusion of left exaxt localizationsgenerated under colimits from small objectslocalization of free cocompletiongenerated under filtered colimits from small objects
(0,1)-category theory(0,1)-toposes$\hookrightarrow$algebraic lattices$\simeq$ Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes$\hookrightarrow$locally presentable categories$\simeq$ Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories$\hookrightarrow$accessible categories
model category theorymodel toposes$\hookrightarrow$combinatorial model categories$\simeq$ Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes$\hookrightarrow$locally presentable (∞,1)-categories$\simeq$
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories$\hookrightarrow$accessible (∞,1)-categories

## References

Secton A.4 and C.2 in

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

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

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

Details are in

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

Revised on November 25, 2013 03:40:00 by Urs Schreiber (89.204.153.236)