Contents

topos theory

# Contents

## Idea

A presheaf on a site is a sheaf if its value on any object of the site is given by its compatible values on any covering of that object.

## Definition

There are several equivalent ways to characterize sheaves. We start with the general but explicit componentwise definition and then discuss more general abstract equivalent reformulations. Finally we give special discussion applicable in various common special cases.

### General definition in components

The following is an explicit component-wise definition of sheaves that is fully general (for instance not assuming that the site has pullbacks).

###### Definition

Let $(C,J)$ be a site in the form of a small category $C$ equipped with a coverage $J$.

A presheaf $A \in PSh(C)$ is a sheaf with respect to $J$ if

• for every covering family $\{p_i : U_i \to U\}_{i \in I}$ in $J$

• and for every compatible family of elements, given by tuples $(s_i \in A(U_i))_{i \in I}$ such that for all $j,k \in I$ and all morphisms $U_j \stackrel{f}{\leftarrow} K \stackrel{g}{\to} U_k$ in $C$ with $p_j \circ f = p_k \circ g$ we have $A(f)(s_j) = A(g)(s_k) \in A(K)$

then

• there is a unique element $s \in A(U)$ such that $A(p_i)(s) = s_i$ for all $i \in I$.
###### Remark

If in the above definition there is at most one such $s$, we say that $A$ is a separated presheaf with respect to $J$.

In this form the definition appears for instance in (Johnstone, def. C2.1.2).-

### General definition abstractly

We now reformulate the above component-wise definition in general abstract terms.

Write

$j : C \hookrightarrow PSh(C)$

for the Yoneda embedding.

###### Definition

Given a covering family $\{f_i : U_i \to U\}$ in $J$, its sieve is the presheaf $S(\{U_i\})$ defined as the coequalizer

$\coprod_{j,k} j(U_j) \times_{j(U)} j(U_k) \stackrel{\overset{}{\to}}{\to} \coprod_i j(U_i) \to S(\{U_i\})$

in $PSh(C)$.

Here the coproduct on the left is over the pullbacks

$\array{ j(U_j) \times_{j(U)} j(U_k) &\stackrel{p_j}{\to}& j(U_j) \\ {}^{\mathllap{p_k}}\downarrow && \downarrow^{\mathrlap{j(f_j)}} \\ j(U_k) &\stackrel{j(f_k)}{\to}& j(U) }$

in $PSh(C)$, and the two morphisms between the coproducts are those induced componentwise by the two projections $p_j, p_k$ in this pullback diagram.

###### Remark

Using that limits and colimits in a category of presheaves are computed objectwise, we find that the sieve $S(\{U_i\})$ defined this way is the presheaf that sends any $K \in C$ to the set of morphisms $K \to U$ in $C$ that factor through one of the $f_i$.

###### Remark

For every covering family there is a canonical morphism

$i_{\{U_i\}} : S(\{U_i\}) \to j(U)$

that is induced by the universal property of the coequalizer from the morphisms $j(f_i) : j(U_i) \to j(U)$ and $j(U_j) \times_{j(U)} j(U_k) \to j(U)$.

###### Definition

A sheaf on $(C,J)$ is a presheaf $A \in PSh(C)$ that is a local object with respect to all $i_{\{U_i\}}$: an object such that for all covering families $\{f_i : U_i \to U\}$ in $J$ we have that the hom-functor $PSh_C(-,A)$ sends the canonical morphisms $i_{\{U_i\}} : S(\{U_i\}) \to j(U)$ to isomorphisms.

$PSh_C(i_{\{U_i\}}, A) : PSh_C(j(U), A) \stackrel{\simeq}{\to} PSh_C(S(\{U_i\}), A) \,.$

Equivalently, using the Yoneda lemma and the fact that the hom-functor $PSh_C(-,A)$ sends colimits to limits, this says that the diagram

$A(U) \to \prod_i A(U_i) \stackrel{\to}{\to} \prod_{j,k} PSh_C(j(U_j) \times_{j(U)} j(U_k), A)$

is an equalizer diagram for each covering family.

This is also called the descent condition for descent along the covering family.

###### Remark

For many examples of sites that appear in practice – but by far not for all – it happens that the pullback presheaves $j(U_j) \times_{j(U)} \times j(U_k)$ are themselves again representable, hence that the pullback $U_j \times_U U_k$ exists already in $C$, even before passing to the Yoneda embedding.

In this special case we may apply the Yoneda lemma once more to deduce

$PSh_C(j(U_j) \times_{j(U)} j(U_k), A) \simeq A(U_j \times_U U_k) \,.$

Then the sheaf condition is that all diagrams

$A(U) \to \prod_i A(U_i) \stackrel{\to}{\to} \prod_{j,k} A(U_j \times_U U_k)$

are equalizer diagrams.

###### Proposition

The condition that $PSh_C(S(\{U_i\}), A)$ is an isomorphism is equivalent to the condition that the set $A(U)$ is isomorphic to the set of matching families $(s_i \in A(U_i))$ as it appears in the above component-wise definition.

###### Proof

We may express the set of natural transformations $PSh_C(j(U_j) \times_{j(U)} j(U_k), A)$ (as described there) by the end

$PSh_C(j(U_j) \times_{j(U)} j(U_k), A) \simeq \int_{K \in C} Set( C(K,U_j) \times_{C(K,U)} C(K,U_k) , A(K)) \,.$

Using this in the expression of the equalizer

$\prod_i A(U_i) \simeq \prod_i \int_{K \in C} Set( C(K,U_i), A(K)) \stackrel{\to}{\to} \prod_{j,k} \int_{K \in C} Set( C(K,U_j) \times_{C(K,U)} C(K,U_k) , A(K))$

as a subset of the product set on the left manifestly yields the componenwise definition above.

###### Definition

A morphism of sheaves is just a morphism of the underlying presheaves. So the category of sheaves $Sh_J(C)$ is the full subcategory of the category of presheaves on the sheaves:

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

### Characterizations over special sites

We discuss equivalent characterizations of sheaves that are applicable if the underlying site enjoys certain special properties.

#### Characterizations over sites of opens

An important special case of sheaves is those over a (0,1)-site such as a category of open subsets $Op(X)$ of a topological space $X$. We consider some equivalent ways of characterizing sheaves among presheaves in such a situation.

(The following was mentioned in Peter LeFanu Lumsdaine’s comment here).

###### Proposition

Suppose $Op = Op(X)$ is the category of open subsets of some topological space regarded as a site with the canonical coverage where $\{U_i \hookrightarrow U\}$ is covering if the union $\cup_i U_i \simeq U$ in $Op$.

Then a presheaf $\mathcal{F}$ on $Op$ is a sheaf precisely if for every complete full subcategory $\mathcal{U} \hookrightarrow Op$, $\mathcal{F}$ takes the colimit in $Op$ over $\mathcal{U} \hookrightarrow Op$ to a limit:

$\mathcal{F}(\underset{\to}{lim} \mathcal{U}) \simeq \underset{\leftarrow}{lim} \mathcal{F}(\mathcal{U}) \,.$
###### Proof

A complete full subcategory $\mathcal{U} \hookrightarrow Op$ is a collection $\{U_i \hookrightarrow X\}$ of open subsets that is closed under forming intersections of subsets. The colimit

$\underset{\to}{\lim} (\mathcal{U} \hookrightarrow Op) \simeq \cup_{i \in I} U_i$

is the union $U \coloneqq \cup_{i \in I} U_i$ of all these open subsets. Notice that by construction the component maps $\{U_i \hookrightarrow U\}$ of the colimit are a covering family of $U$.

Inspection then shows that the limit $\underset{\leftarrow}{\lim}_{i \in I} \mathcal{F}(U_i)$ is the corresponding set of matching families (use the description of limits in terms of products and equalizers ). Hence the statement follows with def. .

#### Characterization over canonical topologies

The above prop. shows that often sheaves are characterized as contravariant functors that take some colimits to limits. This is true in full generality for the following case

###### Proposition

Let $\mathcal{T}$ be be a topos, regarded as a large site when equipped with the canonical topology. Then a presheaf (with values in small sets) on $\mathcal{T}$ is a sheaf precisely if it sends all colimits to limits.

## Sheaves and localization

We now describe the derivation and the detailed description of various aspects of sheaves, the descent condition for sheaves and sheafification, relating it to all the related notions

We start by assuming that a geometric embedding into a presheaf category is given and derive the consequences.

So let $S$ be a small category and write $PSh(S) = PSh_S = [S^{op}, Set]$ for the corresponding topos of presheaves.

Assume then that another topos $Sh(S) = Sh_S$ is given together with a geometric embedding

$f : Sh(S) \to PSh(S)$

i.e. with a full and faithful functor

$f_* : Sh(S) \to PSh(S)$

and a left exact functor

$f^* : PSh(S) \to Sh(S)$

Such that both form a pair of adjoint functors

$f^* \dashv f_*$

with $f^*$ left adjoint to $f_*$.

Write $W$ for the category

$Core(PSh(S)) \hookrightarrow W \hookrightarrow PSh(S)$

consisting of all those morphisms in $PSh(S)$ that are sent to isomorphisms under $f^*$.

$W = (f^*)^{-1}(Core(Sh_S)) \,.$

From the discussion at geometric embedding we know that $Sh(S)$ is equivalent to the full subcategory of $PSh(S)$ on all $W$-local objects.

Recall that an object $A \in PSh(S)$ is called a $W$-local object if for all $p : Y \to X$ in $W$ the morphism

$p^* : PSh_S(X,A) \to PSh_S(Y,A)$

is an isomorphism. This we call the descent condition on presheaves (saying that a presheaf “descends” along $p$ from $Y$ “down to” $X$). Our task is therefore to identify the category $W$, show how it determines and is determed by a Grothendieck topology on $S$ – equipping $S$ with the structure of a site – and characterize the $W$-local objects. These are (up to equivalence of categories) the objects of $Sh$, i.e. the sheaves with respect to the given Grothendieck topology.

###### Lemma

A morphism $Y \to X$ is in $W$ if and only if for every representable presheaf $U$ and every morphism $U\to X$ the pullback $Y \times_X U \to U$ is in $W$

$\array{ Y \times_X U &\to& Y \\ \downarrow^{\in W} && \downarrow^{\Leftrightarrow \in W} \\ U &\to& X } \,.$
###### Proof

Since $W$ is stable under pullback (as described at geometric embedding: simply because $f^*$ preserves finite limits) it is clear that $Y \times_X U \to U$ is in $W$ if $Y \to X$ is.

To get the other direction, use the co-Yoneda lemma to write $X$ as a colimit of representables over the comma category $(Y/const_X)$ (with $Y$ the Yoneda embedding):

$X \simeq colim_{U_i \to X} U_i \,.$

Then pull back $Y \to colim_{U_i \to X} U$ over the entire colimiting cone, so that over each component we have

$\array{ Y \times_X U_i &\to& Y \\ \downarrow && \downarrow \\ U_i &\to& X } \,.$

Using that in $PSh(S)$ colimits are stable under base change we get

$colim_i (Y \times_X U_i) \simeq (colim_i U_i) \times_X Y \,.$

But since $X \simeq colim_i U_i$ the right hand is $X \times_X Y$, which is just $Y$. So $Y = colim_i (Y \times_X U_i)$ and we find that $Y \to X$ is a morphism of colimits. But under $f^*$ the two respective diagrams become isomorphic, since $Y \times_X U_i \to U_i$ is in $W$. That means that the corresponding morphism of colimits $f^*(Y \to X)$ (since $f^*$ preserves colimits) is an isomorphism, which finally means that $Y \to X$ is in $W$.

###### Lemma

A presheaf $A \in PSh(S)$ is a local object with respect to all of $W$ already if it is local with respect to those morphisms in $W$ whose codomain is representable

###### Proof

Rewriting the morphism $Y \to X$ in $W$ in terms of colimits as in the above proof

$\array{ colim_{U \to X} U_i \times_X Y &\stackrel{\simeq}{\to}& Y \\ \downarrow && \downarrow \\ colim_{U \to X} U &\stackrel{\simeq}{\to}& X }$

we find that $A(X) \to A(Y)$ equals

$lim_{U \to X} (A(U) \to A(U \times_X Y)) \,.$

If $A$ is local with respect to morphisms $W$ with representable codomain, then by the above if $Y \to X$ is in $W$ all the morphisms in the limit here are isomorphisms, hence

$\cdots = Id_{A(X)} \,.$
###### Lemma

Every morphism $Y \to X$ in $W \subset PSh(S)$ factors as an epimorphism followed by a monomorphism in $PSh(S)$ with both being morphisms in $W$.

###### Proof

Use factorization through image and coimage, use exactness of $f^*$ to deduce that the factorization exists not only in $PSh(S)$ but even in $W$.

More in detail, given $Y \to X$ we get the diagram

$\array{ Y \times_X Y &&\to&& Y \\ &&& \swarrow \\ \downarrow &&Y \sqcup_{Y \times_X Y} Y && \downarrow \\ & \nearrow && \searrow \\ Y && \to && X } \,.$

Because $f^*$ is exact, the pullbacks and pushouts in this diagram remain such under $f^*$. But since $f^*(Y \to X)$ is an isomorphism by assumption, the all these are pullbacks and pushouts along isomorphisms in $Sh(S)$, so all morphisms in the above diagram map to isomorphisms in $Sh(S)$, hence the entire diagram in $PSh(S)$ is in $W$.

Since the morphism $Y \sqcup_{Y \times_X Y} Y \to X$ out of the coimage is at the same time the equalizing morphism into the image $lim(X \stackrel{\to}{\to} X \sqcup_Y X)$, it is a monomorphism.

###### Definition

The monomorphisms in $PSh(S)$ which are in $W$ are called dense monomorphisms.

###### Lemma

Every monomorphism $Y \to X$ with $X$ representable is of the form

$Y = colim ( U \times_X U \to U )$

for $U = \sqcup_{\alpha} U_\alpha$ a disjoint union of representables

###### Proof

This is a direct consequence of the standard fact that subfunctors are in bijection with sieves.

###### Corollary

If a presheaf $A$ is local with respect to all dense monomorphisms, then it is already local with respect to all morphisms $Y \to X$ of the form

$\array{ Y \\ \downarrow \\ X } = colim \left( \array{ W &\stackrel{\to}{\to}& U \\ \;\;\downarrow^{dense mono} && \downarrow^{Id} \\ U \times_X U & \stackrel{\to}{\to}& U } \right)$

with the left vertical morphism a dense monomorphism

(and with $U = \sqcup_\alpha U_\alpha$ the disjoint union (of representable presheaves) over a covering family of objects.)

###### Definition

The morphisms in $W$ with representable codomain

• of the form $colim (U \times_X U \stackrel{\to}{\to} U) \to X$ as above are covers:

• of the form $colim (W \stackrel{\to}{\to} U) \to X$ (with $W$ a cover of $U \times_X U$) as above are hypercovers

of the representable $X$.

###### Proposition

A presheaf $A$ is $W$-local, i.e. a sheaf, already if it is local (satisfies descent) with respect to all covers, i.e. all dense monomorphisms with codomain a representable.

Urs: the above shows this almost. I am not sure yet how to see the remaining bit directly, without making recourse to the full machinery leading up to section VII, 4, corollary 7 in Sheaves in Geometry and Logic.

So we finally conclude:

###### Corollaries

We have:

• Systems $W$ of weak equivalences defined by choice of geometric embedding $f : Sh(S) \to PSh(S)$ are in canonical bijection with choice of Grothendieck topology.

• A presheaf $A$ is $W$-local, i.e. local with respect to all local isomorphisms, if and only if it is local already with respect to all dense monomorphism, i.e. if and only if it satisfies sheaf condition for all covering sieves.

From the assumption that $f : Sh(S) \to PSh(S)$ is a geometric embedding follows at once the following explicit description of the sheafification functor $f^* : PSh(S) \to Sh(S)$.

###### Lemma (Sheafification)

For $A \in PSh(S)$ a presheaf, its sheafification $\bar A := f_* f^* A$ is the presheaf given by

$\bar A : U \mapsto colim_{(Y \to U) \in W} A(Y)$
###### Proof

By the discussion at geometric embedding the category $Sh(S)$ is equivalent to the localization $PSh(S)[W^{-1}]$, which in turn is the category with the same objects as $PSh(S)$ and with morphisms given by spans out of hypercovers in $W$

$PSh(S)[W^{-1}](X,A) = colim_{(Y \to X) \in W} A(Y) \,.$

So we have

$\array { Sh(S) &&\stackrel{\stackrel{f_*}{\to}}{\stackrel{f^*}{\leftarrow}}& PSh(S) \\ & \searrow_{\simeq}&\Downarrow^{\simeq}& \downarrow \\ && PSh(S)[W^{-1}] \,. }$

and deduce

• by Yoneda that $\bar A(U) = PSh_S(U, \bar A)$;

• by the hom-adjunction this is $\cdots \simeq Sh_S(\bar U, \bar A)$;

• by the equivalence just mentioned this is $\cdots \simeq PSh_S[W^{-1}](U,A)$.

###### Remark: covers versus hypercovers

For checking the sheaf condition the dense monomorphisms, i.e. the ordinary covers are already sufficient. But for sheafification one really needs the local isomorphisms, i.e. the hypercovers. If one takes the colimit in the sheafification prescription above only over covers, one obtains instead of sheafification the plus-construction.

###### Definition: plus-construction

For $A \in PSh(S)$ a presheaf, the plus-construction on $A$ is the presheaf

$A^+ : X \mapsto colim_{(Y \hookrightarrow X) \in W } A(Y)$

where the colimit is over all dense monomorphisms (instead of over all local isomorphisms as for sheafification $\bar A$).

###### Remark: plus-construction versus sheafification

In general $A^+$ is not yet a sheaf. It is however in general closer to being a sheaf than $A$ is, namely it is a separated presheaf.

But applying the plus-construction twice yields the desired sheaf

$(A^+)^+ = \bar A \,.$

This is essentially due to the fact that in the context of ordinary sheaves discussed here, all hypercovers are already of the form

$colim(W \stackrel{\to}{\to} U)$

for $W \to U \times_X U$ a cover. For higher stacks the hypercover is in general a longer simplicial object of covers and accordingly if one restricts to covers instead of using hypercovers one will need to use the plus-construction more and more often. Specifically, for stacks of $n$-groupoids one needs to apply the plus-construction $n+2$ times; see plus construction on presheaves.

When $n=\infty$, even a countable sequence of applications does not suffice in general, but a sufficiently long transfinite sequence does. In this case, using hypercovers instead actually produces a different answer, namely the reflection into the hypercompletion of the sheaf $\infty$-topos.

## Examples

The archetypical example of sheaves are sheaves of functions:

• for $X$ a topological space, $\mathbb{C}$ a topological space and $O(X)$ the site of open subsets of $X$, the assignment $U \mapsto C(U,\mathbb{C})$ of continuous functions from $U$ to $\mathbb{C}$ for every open subset $U \subset X$ is a sheaf on $O(X)$.

• for $X$ a complex manifold and $\mathbb{C}$ a complex manifold, the assignment $U \mapsto C_{hol}{X,\mathbb{C}}$ of holomorphic functions in a sheaf.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-$\infty$-groupoid

The original definition is in

• Jean Leray, L’anneau d’homologie d’une représentation. Comptes rendus hebdomadaires des séances de l’Académie des Sciences 222 (1946), 1366–1368. PDF

Section C2 in

A concise and contemporary overview can be found in

• C. Centazzo, E. M. Vitale, Sheaf theory , pp.311-358 in Pedicchio, Tholen (eds.), Categorical Foundations , Cambridge UP 2004. (draft)

The book by Kashiwara and Schapira discusses sheaves with motivation from homological algebra, abelian sheaf cohomology and homotopy theory, leading over in the last chapter to the notion of stack.

A quick pedagogical introduction with an eye towards the generalization to (∞,1)-sheaves is in

Classics of sheaf theory on topological spaces are

• Roger Godement, Topologie algébrique et théorie des faisceaux, Hermann, 1958, 283 p. gBooks

Recently, an improvement in understanding the interplay of derived functors (inverse image and proper direct image) in sheaf theory on topological spaces has been exhibited in

• Olaf M. Schnuerer, Wolfgang Sergel, Proper base change for separated locally proper maps, arxiv/1404.7630