Schreiber sheaves

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• in the previous bit we had worked out the detailed description of geometric embeddings $f : F \hookrightarrow E$ of categories with finite limits (or of topoi). We found three equivalent characterizations:

  • $F \hookrightarrow E$ is a geometric embedding;

  • $F$ is (equivalent to) the full subcategory on $W$-local objects;

  • $F$ is (equivalent to) the localization $E[W^{-1}]$.

• we now apply this to the case that $E = PSh(S)$ is a category of presheaves and discover this way the notions

  • coverage– a characterization of the weak equivalences $W$ in $E = PSh(S)$

  • sheaf– the $W$-local objects

• the formalism is set up in such a way that from there we arrive with ease at

  • the description of sheafification;

  • the generalization to (the homotopy category) of infinity-stacks

  • abelian sheaf cohomology

Geometric embeddings into presheaf topoi

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 $f^*$.

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

In the present situation of presheaves, we call the morphisms in $W$ local isomorphism.

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 determined by a coverage or 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 $Sh(S)$, 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 $h : Y \to X$ we get the diagram

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

The outer square is a pullback, the inner a pushout.

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.

Sieves

Lemma

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

$$Y = colim ( U \times_X U \stackrel{\to}{\to} U )$$

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.

Sieves are an equivalent way to encode subobjects of representable functors in a presheaf category in terms of the total sets of elements of such a subfunctor.

The notion of sieve is usually used when certain such subobjects are singled out as cover?s of a coverage: the singled out subobjects then correspond to covering sieves.

Let $C$ be a small category.

Definition

A sieve $S$ (Fr. crible) on an object $c \in C$ is a subset $S \subset Ob(C/c)$ of the set of objects of the over category over $c$ which is closed under precomposition: it has the property that with $(d \to c) \in S$ for every morphism $(e \to d) \in Mor(C)$ also the composite $(e \to d \to c)$ is in $S$.

Remark

Sometimes the condition of a sieve being closed under the operation of precomposing with an arbitrary morphism $g: e \to d$ is called a “saturation condition”. Given any collection of morphisms targeted at $c$, one can always close it up or saturate it, to obtain a sieve on $c$.

There is a canonical way to create subfunctors from sieves and sieves from subfunctors.

Definition

Given a sieve $S$ on $c$, the subfunctor $F_S \hookrightarrow Y(c)$ defined by the sieve is the presheaf

• that assigs to each object $d \in C$ the set $F_S(d) = \{(d \to c) \in S\}$;

• that assigns to each morphism $(d \to d') \in C$ the function $F_S(d') \to F_S(d)$ induced on elements by precomposition with $d \to d'$.

Definition

Given a subfunctor $F \hookrightarrow Y(c)$, the sieve defined by the subfunctor is given by

• $S_F := \{g: d \to c \in Mor(C) | Y(d) \stackrel{Y(g)}\to Y(c) = Y(d) \to F \to Y(c) \}$

or equivalently

• $S_F = \coprod_{d \in Obj(C)}F(d)$.

Lemma

These two definitions establish a bijection between sieves on $c$ and subobjects of $Y(c)$.

For every sieve $S$ we have

$$S_{F_S} = S$$

and for every subfunctor we have

$$F_{S_F} = F \,.$$

Remarks

• The construction of $S_F$ makes sense for every morphism of presheaves $F \to Y(c)$. The sieve is sensitive precisely to the image of this map,

  $$S_{F} = S_{im (F \to Y(c))} \,.$$

• In the presence of a coverage a morphism $F \to Y(c)$ is sometimes called a local epimorphism if the sieve $S_F$ is a covering sieve. If $F \to Y(c)$ is actually a subfunctor, then it is called a dense monomorphism.

• The pullback of a subfunctor $i: F_S \hookrightarrow Y(c) = \hom(-, c)$ along any morphism $\hom(-, g): \hom(-, d) \to \hom(-, c)$ is again a subfunctor $g^* F$ of $d$, hence sieves are closed under pulling back. Concretely,

  $$g^* S = \bigcup_{e \in Ob(C)} \{f: e \to d: g f \in S\} \,.$$

• A sieve $S_F$ on $c$, for $i: F \hookrightarrow \hom(-, c)$ a subfunctor, may be described as a function which assigns to each object $d$ a collection of morphisms $f: d \to c$ into $c$. Naturality of the inclusion $i$ means that whenever $f: d \to c$ belongs to the sieve and $g: e \to d$ is any morphism, then $f g: e \to c$ also belongs to the sieve.

Lemma

The subfunctor $F_S \hookrightarrow X$ corresponding to a sieve $S$ is the coimage of the morphism out of the disjoint union of all objects (regarded as representable presheaves) in the sieve:

$$(U = \coprod_{(U_\alpha \to X) \in S} U_\alpha) \to X$$

in that

$$(F_S \to X) = ((colim(U \times_X U \stackrel{\to}{\to} U) \to X) \,.$$

If the sieve is generated by (is the saturation of) a collection of morphisms $\{U_\alpha \to U\}$ then the same statement remains true with $U$ being the coproduct over just these $U_\alpha$.

Proof

As described at limits and colimits by example, the colimit of presheaves may be computed objectwise in $Set$. Doing so and using the Yoneda lemma tells us that for each object $V$ we have

$$\begin{aligned} colim(U \times_X U \stackrel{\to}{\to} U)(V) & \simeq colim(Hom(V,U \times_X U) \stackrel{\to}{\to} Hom(V,U)) \\ & \simeq colim( Hom(V,U) \times_{Hom(U,X)} Hom(V,U) \stackrel{\to}{\to} Hom(V,U)) \end{aligned}$$

where all objects appearing (at least in the first lines) are implicitly regarded as presheaves under the Yoneda embedding.

But this colimit now manifestly computes the set

$$\{ maps from V to X that factor through U\}_\sim$$

where the equivalence relation is

$$((f \to U \to X) \sim (g \to U \to X)) \Leftrightarrow (f and g coincide as maps to X) \,.$$

So the set is just the set of maps from $V$ to $X$ that factor through one of the $U_\alpha$, which is precisely the set $F_S(V)$ assigned by the subfunctor corresponding to the sieve.

Examples

• For $X$ a topological space let $Op(X)$ be the category of open subsets of $X$ and consider presheaves $PSh(X) := [Op(X)^{op}, Set]$ on $X$. For any open subset $c = V \in Op(X)$ let $\{d_i\} = \{U_i\}$ be a cover of $V$ by open subsets $U_i$ in the ordinary sense (i.e. each $U_i$ is an open subset of $V$ and their joint union is $V$, $\bigcup_i U_i = V$), then $\pi : (\coprod_i Y(U_i)) \stackrel{\coprod_i U_i \hookrightarrow_i X}{\to} Y(V)$ (with $Y$ the Yoneda embedding) is a local epimorphism of presheaves on $V$ and its image – or equivalently its coimage – is the subfunctor $(F := \bigcup_i Y(U_i)) \hookrightarrow Y(V)$ that sends each $W \in Op(X)$ to the set of maps $W \to V$ that factor through one of the $U_i$. The collection of all such maps for all choice of $W$ is the corresponding covering sieve $\{ f : W \to V \in Mor(S) \;|\; f = W \to U_i \to V \}$.

• The situation for more general sites $S$ other than $Op(X)$ is literally the same as above, with $U_i, W, V$ etc. objects of $S$.

A detailed description of what’s going on

The followig is a pedagogical step-by-step description of the crucial aspects of sieves as covers.

To start with the simplest example that already contains in it all the relevant aspects, consider a topological space $X$ with an open subset $V \subset X$ that is covered by two open subsets $U_1, U_2 \subset X$ in that the union $U_1 \cup U_2$ in $X$ coincides with $V$:

$$U_1 \cup U_2 = V \,.$$

This is the coproduct in the category of open subsets of $X$. Although it's not a disjoint coproduct, which coproducts in sheaf categories are, to which we come.

Another way to think of this is obtained by first forming the fiber product of $U_1$ with $U_2$ over $V$ in $Op(X)$, which is the intersection $U_1 \cap U_2$ sitting in the pullback diagram

$$\array{ U_1 \times_V U_2 &\to& U_1 \\ \downarrow && \downarrow \\ U_2 &\to& V }$$

in $Op(X)$.

The union $U_1 \cup U_2$ can be obtained from this by removing in the above diagram the bottom right corner and then forming the pushout over the resulting diagram: this is again $V$, i.e. the diagram

$$\array{ U_1 \times_V U_2 &\to& U_1 \\ \downarrow && \downarrow \\ U_2 &\to& U_1 \cup U_2 = V }$$

is not only a pullback also a pushout diagram.

The important point about (covering) sieves is that they show up when the above situation is sent via the Yoneda embedding from $Op(X)$ to presheaves on $Op(X)$. The crucial aspect here that gives rise to the peculiarities of sieves is that

• the Yoneda embedding preserves limits, but not, in general, colimits

As a result, the above discussion goes through equivalently for the presheaves represented by our open subsets all the way up to the last pushout. In $Op(X)$ that last pushout reproduced the open subset $V$. In $PSh(X) = [Op(X)^{op}, Set]$ it instead reproduces the sieve on $V$ generated by $U_1$ and $U_2$.

Let’s go through this in detail. First of all notice that in $PSh(S)$ all limits and colimits do exist (see limits and colimits by example for more on that), so that for instance the coproduct

$$Y(U_1) \sqcup Y(U_2)$$

does always exist (as opposed to its would-be cousin $U_1 \sqcup U_2$). Here $Y$ denotes the Yoneda embedding which we here indicate explicitly, even though often and elsewhere, notably elsewhere in this entry here, it is notationally suppressed.

For the following it is helpful to say explicitly what the presheaf $Y(U_1) \sqcup Y(U_2)$ is like. Since, as described at limits and colimits by example, colimits of presheaves are computed objectwise, we know that this presheaf evaluated on any open set $W \subset X$ yields the set

$$\begin{aligned} (Y(U_1) \sqcup Y(U_2))(W) &= Y(U_1)(W) \sqcup Y(U_2)(W) \\ &= Hom(W,U_1) \sqcup Hom(W, U_2) \end{aligned} \,,$$

where the coproducts on the right are just those in Set which are just ordinary disjoint unions of sets.

So this says that $Y(U_1) \sqcup Y(U_2)$ is the presheaf that assigns to any open set $W$ the dijoint union of the collections of maps from $W$ to $U_1$ and those from $W$ to $U_2$ in $X$. (Since $Op(X)$ is a poset there is either none or one such map in each case, but it is helpful to speak generally of “sets of all maps”, since that is the general intuition useful for presheaf categories. $Op(X)$ just happens to be a particularly simple example.)

Notice that in particular a given map $W \to V$ which factors both through $U_1 \to V$ as well as through $U_2 \to V$ will appear as two distinct elements in the set $(Y(U_1) \sqcup Y(U_2))(W)$. This we’ll come back to in a minute.

But first consider the fiber product from before, now after having applied the Yoneda embedding. Since we know from general nonsense that this preserves fiber products, we know that the pullback presheaf $Y(U_1) \times_{Y(V)} Y(U_2)$ in

$$\array{ Y(U_1) \times_{Y(V)} Y(U_2) &\to& Y(U_2) \\ \downarrow && \downarrow \\ Y(U_1) &\to& Y(V) }$$

is the same as $Y(U_1 \times_V U_2) = Y(U_1 \cap U_2)$.

But this is also easily checked explicitly. We go through this because this kind of reasoning for computing limits and colimits of presheaves will be needed throughout here: since for any $W$ the covariant hom-functor $Psh(Y(W),-) : Psh \to PSh$ preserves limits (by the very definition of limit!) we have for every $W$ a pullback diagram of sets

$$\array{ Hom(Y(W),Y(U_1) \times_{Y(V)} Y(U_2)) &\to& Hom(Y(W),Y(U_2)) \\ \downarrow && \downarrow \\ Hom(Y(W),Y(U_1)) &\to& Hom(Y(W),Y(V)) }$$

Again by the Yoneda lemma this is simply

$$\array{ (Y(U_1) \times_{Y(V)} Y(U_2) )(W) &\to& Hom(W,U_2) \\ \downarrow && \downarrow \\ Hom(W,U_1) &\to& Hom(W,V) } \,.$$

This being a pullback diagram now says in words:

The set $(Y(U_1) \times_{Y(V)} Y(U_2) )(W)$ is the set of those pairs of maps $W \to U_1$ and $W \to U_2$ that coincide as maps $W \to U_1 \to V$ and $W \to U_2 \to V$ to $V$.

Clearly, this set is the same as the set of maps into the intersection $U_1 \cap U_2$, so indeed

$$Y(U_1) \times_{Y(V)} Y(U_2) = Y(U_1 \times_V U_2) = Y(U_1 \cap U_2) \,.$$

So far so long-winded. Now let’s see what happens when we now form the pushout over

$$\array{ Y(U_1 \times_V U_2) &\to& Y(U_2) \\ \downarrow \\ Y(U_1) }$$

that will go, for a moment, by its canonical but lenghty name $Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2)$

$$\array{ Y(U_1 \times_V U_2) &\to& Y(U_2) \\ \downarrow && \downarrow \\ Y(U_1) &\to& Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2) } \,.$$

Again, we can figure out what this presheaf is by computing objectwise what it does to any open subset $W$: since colimits of presheaves are computed objectwise, the diagram

$$\array{ Hom(W, U_1 \times_V U_2) &\to& Hom(W, U_2) \\ \downarrow && \downarrow \\ Hom(W,U_1) &\to& (Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2))(W) }$$

must be a colimit in Set. Again, this is easily read out in words:

The set $(Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2))(W)$ is the quotient of the disjoint union of the collection of maps from $W$ into $U_1$ and those from $W$ into $U_2$, by the equivalence relation which identifies two such maps $W \to U_1$ and $W \to U_2$ if they both factor through a map $W \to U_1 \times_X U_2$, i.e. if they both land in the intersection $U_1 \cap U_2$ and coincide there.

But this just means that contrary to the plain coproduct $Y(U_1) \sqcup Y(U_2)$, two maps $W \to U_1$ and $W \to U_2$ that coincide as maps $W \to X$ are no longer regarded as different elements of our set given by the pushout presheaf, but are regarded as being the same.

So this means we find that

$$(Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2))(W) = \{ maps W \to V that factor through either U_1 or U_2 \} \,.$$

But this is by definition the assignment of the subfunctor coresponding to the sieve on $V$ generated by $U_1 \to V$ and $U_2 \to V$.

So we find that

$$F_{sieve(U_1,U_2)} = Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2) \,.$$

Given that we made it to this point, we should go one small step further that will be very useful.

In the present simple example we worked with a cover given by just two objects $U_1$ and $U_2$. Of course in general the cover will consist of more than just two objects. Then the above kind of notation becomes a bit cumbersome. But there is a simple reformulation that makes everything look nice again.

Namely, let’s come back to the observation that the coproduct $Y(U_1) \sqcup Y(U_2)$ does exist. Let’s just call this presheaf $\mathbf{U}$. (not in general a representable?!).

Then it is easy to see by the same kind of objectwise reasoning that the colimiting presheaf that we are after is equivalently the colimit over the pair of parallel morphisms

$$\mathbf{U} \times_{Y(V)} \mathbf{U} \stackrel{\stackrel{p_2}{\to}}{\stackrel{p_1}{\to}} \mathbf{U}$$

in that

$$F_{sieve(U_1,U_2)} \simeq colim ( \mathbf{U} \times_{Y(V)} \mathbf{U} \stackrel{\stackrel{p_2}{\to}}{\stackrel{p_1}{\to}} \mathbf{U} ) \,.$$

This description now has an evident direct generalization to the case where instead of just $U_1 \to V$ and $U_2 \to V$ we have an arbitrary collection $\{U_i \to V\}$ of open sets $U$ covering $V$. One finds again with

$$\mathbf{U} := \coprod_i Y(U(i))$$

that

$$F_{sieve(\{U_i\})} \simeq colim ( \mathbf{U} \times_{Y(V)} \mathbf{U} \stackrel{\stackrel{p_2}{\to}}{\stackrel{p_1}{\to}} \mathbf{U} )$$

is the presheaf that to every $W$ assigns the set of all maps $W \to V$ that factor through any one of the $U_i$.

It is in this way that sieves and their associated subfunctors encode the notion of cover of an object $V$: they tell us which of all the maps into $V$ do factor through the cover.

And, to end this pedagocial piece with an outlook to indicate the gain in understanding this achieves:

once we start forming $\mathbf{U} \times_{Y(V)} \mathbf{U} \stackrel{\to}{\to} \mathbf{U}$ there is no stopping. We can keep forming higher and higher such fiber products

$$\cdots \mathbf{U} \times_{Y(V)} \mathbf{U} \times_{Y(V)} \mathbf{U} \times_{Y(V)} \mathbf{U} \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}} \mathbf{U} \times_{Y(V)} \mathbf{U} \times_{Y(V)} \mathbf{U} \stackrel{\to}{\stackrel{\to}{\to}} \mathbf{U} \times_{Y(V)} \mathbf{U} \stackrel{\to}{\to} \mathbf{U} \,.$$

When one passes from just presheaves to (infinity,1)-presheaves, then covering presheaves will be given by the right kind of colimit over these simplicial diagrams (namely the homotopy colimit). More on that is at descent.

Sheaves

Let $F_S \to X$ be a dense monomorphism coming from a sieve $S = \{U_\alpha \to X\}$.

The condition that a presheaf $A$ be local with respect to $F_S \to X$ is called the sheaf condition. We now spell out in more detail what this condition says.

sheaf condition for presheaves on open subsets

From the above detailed discuss recall that $F_{sieve(\{U_i\})}$ is precisely the coequalizer? of the obvious pair of morphisms

$$\coprod_{i, j} \hom(-, U_i \cap U_j) \stackrel{\to}{\to} \coprod_i hom(-, U_i)$$

with $hom(-, U_i) := Y(U_i)$ denoting the presheaf represented? under the Yoneda embedding? by $U_i$, as usual.

Here the domain of this parallel pair is the pullback? of the evident map $\coprod_i hom(-, U_i) \to hom(-,X)$
along itself, and the two parallel arrows are the projection maps out of this pullback:

$$\array{ \coprod_{i,j} hom(-,U_i \cap U_j) &\to& \coprod_i hom(-, U_j) \\ \downarrow && \downarrow \\ \coprod_i hom(-, U_i) &\to& hom(-,X) }$$

Thus for $G$ any presheaf?, maps $F \to G$ are precisely the same as maps $\coprod_i \hom(-, U_i) \to G$ which coequalize the parallel pair. Applying $Hom_{Set^{C^{op}}}(-,G)$ to the colimit diagram

$$\coprod_{i, j} \hom(-, U_i \cap U_j) \stackrel{\to}{\to} \coprod_i hom(-, U_i) \to F$$

yields the limit diagram

$$Hom(F, G) \to Hom(\coprod_i hom(-, U_i), G) \stackrel{\to}{\to} Hom(\coprod_{i, j} \hom(-, U_i \cap U_j), G)$$

which using Yoneda? is the equalizer diagram

$$Hom(F,G) \to \prod_i G(U_i) \stackrel{\to}{\to} \prod_{i, j} G(U_i \cap U_j)$$

and hence identifies $Hom(F,G)$ indeed as the set of descent? data for the sheaf? condition on $G$.

sheaf condition more generally

The above generalized essentially literally to sites that have pullbacks. For instance the canonical example of a category of subsets does: pullbacks are just intersections $U_\alpha \times_X U_\beta = U_\alpha \cap U_\beta$ of open subsets.

Using the Yoneda lemma and the fact that the contravariant Hom takes colimits to limits we find that

$$(Hom(X,A) \to Hom(colim(U \times_X U \stackrel{\to}{\to} U), A)) = (A(X) \to lim ( \prod_\alpha A(U_\alpha) \stackrel{\to}{\to} \prod_{\alpha, \beta} A(U_\alpha \times_X U_\beta)) )$$

So the sheaf condition on $A$ says that for all covering sieves on a representable $X$, $A(X)$ has to be the above limit (an equalizer).

By the general prescription this limit is precisely:

• that subset whose elements have the property that restricted to double intersections they coin

$$\{(\sigma_\alpha \in A(U_\alpha))| \forall \alpha, \beta : \sigma_\alpha|_{U_{\alpha, \beta}} = \sigma_\beta|_{U_\alpha, \beta}\} \subset \prod_\alpha A(U_\alpha)$$

Now that we understand the locality condition with respect to dense monomorphisms, we continue with the discussion of the general characterization of $W$-local objects. It will turn out that locality with respect to dense monomorphisms already implies $W$-locality. But the general local isomorphisms are still important, for instance for sheafification, described further below.

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.

Remark

The above shows this almost. The claim follows using the full machinery leading up to section VII, 4, corollary 7 in Sheaves in Geometry and Logic, which we shall not go through here.

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.

Coverages

We have seen that only the sieves corresponding to the dense monomorphisms in $W$ matter for the definition of sheaves. Therefore often geometric embedding $f : PSh(S) \to Sh(S)$ is characterized by this collection of sieves.

A coverage on a category $C$ is a collection of families of coterminal morphisms $\{f_i:U_i\to U\}_{i\in I}$ to be thought of as covering families. The essential characteristic of these covering families is that they be “stable under pullback.” A number of other “saturation” conditions are frequently also imposed for convenience. A category equipped with a coverage is called a site.

One of the main purposes of a coverage is that it provides the minimum structure necessary to define a notion of sheaf (or more generally stack) on $C$. A Grothendieck topos is defined to be the category of sheaves (of sets) on a small site. From this perspective, the example to keep in mind is the poset $O(X)$ of open sets in some topological space (or locale) $X$, where a morphism is an inclusion, and a family of inclusions $\{U_i \hookrightarrow U\}$ is a covering family iff $U = \bigcup_i U_i$.

Another perspective on a coverage is that the covering families are “postulated well-behaved quotients.” That is, saying that $\{f_i:U_i\to U\}_{i\in I}$ is a covering family means that we want to think of $U$ as a well-behaved quotient (i.e. colimit) of the $U_i$. Here “well-behaved” means primarily “stable under pullback.” In general, $U$ may or may not actually be a colimit of the $U_i$; if it always is we call the site subcanonical. From this perspective, the embedding of $C$ into its category of sheaves is “the free cocompletion of $C$ that takes covering families to well-behaved quotients”; compare how the Yoneda embedding of an arbitrary category $C$ into its category of presheaves is its free cocompletion, period.

The traditional name for a coverage, with the extra saturation conditions imposed, is a Grothendieck topology, and this is still widely used in mathematics. Following Sketches of an Elephant, on this page we use coverage for a pullback-stable system of covering families and Grothendieck coverage if the extra saturation conditions are imposed. See Grothendieck topology for a discussion of the objections to that term.

Definition

A coverage on a category $C$ consists of a collection of families of coterminal morphisms $\{f_i:U_i\to U\}_{i\in I}$, called covering families, such that

• If $\{f_i:U_i\to U\}_{i\in I}$ is a covering family and $g:V\to U$ is a morphism, then there exists a covering family $\{h_j:V_j\to V\}$ such that each composite $g h_j$ factors through some $f_i$.

Definition

A site is a category equipped with a coverage. Often sites are required to be small (see large site).

Sheafification

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

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

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 $W$

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

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 $\cdots \simeq PSh_S[W^{-1}](U,A)$.

Covers and hypercovers

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 howver in general closer to being a sheaf than $A$ is, in that 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.