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The notion of sieve is a generalization of that of (right) ideal in a monoid from monoids to categories: a sieve on an object $X$ in a category $\mathcal{C}$ is a collection of morphisms with codomain $X$ that are closed under precomposition with morphisms in $\mathcal{C}$.
Sometimes one says that a sieve in a category $\mathcal{C}$ is a full subcategory closed under precomposition with morphisms in $\mathcal{C}$ (Lurie, def. 6.2.2.1). If so, then a sieve on an object $X$ is a sieve in the slice category $\mathcal{C}_{/X}$. But a sieve in a category is also naturally taken to be a collection of sieves on various objects. On the other hand, most authors speak just about sieves on an object anyway.
Sieves on objects are an equivalent way to talk about 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 covers of a (sifted) coverage: the singled out subobjects then correspond to covering sieves. Accordingly, the classical examples of sieves on an object are Grothendieck topologies, used to present the presheaves that behave like coverings. They are used to say which presheaves are actually sheaves with respect to a given coverage or Grothendieck topology.
A choice of collections of morphism $d \to c$ into an object $c \in C$ for each $d \in C$ reminds one of the representable functor presheaf $Y(c) := \hom_C(-, c): C^{op} \to Set$ which assigns to each $d \in C$ the set of all morphisms from $d$ to $c$. Every choice of covers of $c$ is therefore for each $d \in C$ a subset of the value of this functor evaluated at $d$. This begins to look like an monic natural transformation into this functor. Indeed, it turns out that one can assume without restriction of generality that the assignment of covers of $c$ can always be extended to such a monic natural transformation, and hence one formalizes the notion of βcollection of coversβ of $c$ as a subobject $i: F \hookrightarrow \hom(-, c)$ of the functor represented by $c$: a sieve at $c$.
A dual notion is a cosieve.
Let $C$ be a small category.
A sieve $S$ in $C$ is a functor $S \hookrightarrow C$ that is both fully faithful and a discrete fibration.
A sieve on an object $c \in C$ is a sieve in the slice category $C/c$.
Spelling this out, we arrive at the traditional definition of a sieve on an object:
A sieve $S$ 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 whenever $(d \to c) \in S$ and $(e \to d) \in Mor(C)$ then the composition $(e \to d \to c)$ is in $S$.
This is probably called a sieve because it βsifts outβ the βspecialβ maps into $c$ from the set of all maps into $c$. (Note that βsieveβ is the noun, while βsiftβ is the verb.)
The French term for a sieve is crible.
Sometimes the condition of a sieve being closed under the operation of precomposing with an arbitrary morphism $g: e \to d$ is called a βsaturation conditionβ. 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.
A subfunctor is a subobject in a functor category. Here, specifically, one is interested in subobjects in a presheaf category of representable functors. Itβs these subfunctors of representable functors that are in bijection with sieves.
Given a sieve $S$ on $c$, the subfunctor $F_S \hookrightarrow Y(c)$ defined by the sieve is the presheaf
that assigns 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 $d \to d'$.
Given a subfunctor $F \hookrightarrow Y(c)$, the sieve defined by the subfunctor is given by
or equivalently
These two definitions establish a bijection between sieves on $c$ and subobjects of $Y(c)$.
For every sieve $S$ we have
and for every subfunctor we have
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,
In the presence of a Grothendieck topology 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,
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.
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:
in that
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 $U_\alpha$.
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
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
where the equivalence relation is
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.
The following 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 $V$:
This is the coproduct in the category of open subsets of $X$, but that behaves very differently from the disjoint coproducts that we are used to from categories like Set. On the other hand, when one comes to a category of presheaves or sheaves (a Grothendieck topos), this topos does have disjoint coproducts.
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
in $Op(X)$.
The union $U_1 \cup U_2$ is again 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
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
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 $U_2$.
Letβs go through this in detail. First of all notice that in $PSh(X)$ all limits and colimits do exist (see limits and colimits by example for more on that); in particular the coproduct
exists. 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 $Y(U_1) \amalg Y(U_2)$ 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
where the coproducts on the right are just those in Set which are just ordinary disjoint unions of sets.
So this says that $Y(U_1) \amalg Y(U_2)$ is the presheaf that assigns to any open set $W$ the disjoint 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 βsets of all mapsβ, 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 $(Y(U_1) \amalg Y(U_2))(W)$. This weβll 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
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
Again by the Yoneda lemma this is simply
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
So far so long-winded. Now letβs see what happens when we now form the pushout over
that will go, for a moment, by its canonical but lenghty name $Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2)$
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
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 $Y(U_1) \amalg Y(U_2)$, 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
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
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, letβs come back to the observation that the coproduct $Y(U_1) \amalg Y(U_2)$ does exist. Letβs 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
in that
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
that
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 encapsulate 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
When one passes from just presheaves to (β,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.
We now show that the subfunctor $F_S$ associated with a sieve $S$ coming from a cover $\{U_i \to X\}$ is the right kind of morphism to require a sheaf to be a local object for, by demonstrating that using it the usual sheaf condition on a presheaf with respect to the cover $\coprod_i U_i$ is reproduced:
From the above detailed discuss recall that $F_{sieve(\{U_i\})}$ is precisely the coequalizer of the obvious pair of morphisms
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:
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
yields the limit diagram
which using Yoneda is the equalizer diagram
and hence identifies $Hom(F,G)$ indeed as the set of descent data for the sheaf condition on $G$.
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 β or equivalently its coimage β 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 standard textbook account on sieves in category theory is in
In the context of (β,1)-categories sieves are discussed around def. 6.2.2.1 of