previous: sheaves
home: sheaves and stacks
next: infinity-groupoids
we had defined categories of sheaves as geometric embeddings into presheaf categories $Sh(C) \hookrightarrow PSh(C)$.
and then characterized sheaves $F$ as presheaves that are local with respect to morphisms induced by covering sieves $\{U_i \to V\}$
the standard example to keep in mind in the following is
$C = Op(X)$ is the category of open subsets of a manifold $X$,
a covering sieve $\{U_i \to V\}$ is precisely a sieve generated by an ordinary open cover of $V by the \{U_i\}$
a typical example of a sheaf is the sheaf of differential $n$-forms $\Omega^n(X)$
as a preparation for our next step – the definition of infinity-stacks – we now look at geometric morphisms between such categories of sheaves $Sh(X) \to Sh(Y)$.
in particular we will characterize points of sheaf topoi in the guise of geometric morphisms
it will turn out that a morphism $F : A \to B$ of sheaves is an isomorphism precisely if its inverse image $x^* F : x^* A \to x^* B$ is an isomorphism of sets
From now on we concentrate for simplicitly and convenience on sheaves on topological spaces, i.e. on categories of open subsets of topological spaces.
We now characterize isomorphisms of sheaves as those transformations that are isomorphisms of sets locally over each point of $X$, namely stalkwise.
This is to prepare the ground for the natural generalization to infinity-stacks: an $\infty$-stack will be modeled by a sheaf of simplicial sets, such that a weak equivalence between two such simplicial sheaves is a transformation which is on each stalk a weak equivalence of simplicial sets.
A site is a category $C$ equipped with a coverage? $J$, i.e. with a choice of covering sieves among all sieves.
Following our previous result we call $Sh_J(X)$ the homotopy category $Ho_J(PSh(C))$ of presheaves on $C$ with respect to the monomorphisms defined by the covering sieves. The collection $W$ of morphism that are sent to isomorphisms under $PSh(C) \to Sh_J(C)$ are the local isomorphisms defined by the site.
Motivated from the archetypical example of categories of open subsets?, one says that
a presite is the same as a category? $S$;
a morphism of presites $S \to S'$ is a functor? going the other way round, $S' \to S$.
Hence the category of presites is just the opposite category? of the category Cat? of categoris,
This is just terminology, but supposedly suggestive for working with presheaf categories. For instance one would denote a presite by $X$ and the same entity regarded as a category as $S_X$ and write
and thus have notation entirely analogous to the familiar notation for presheaves on a topological space $X$.
In this notation a site $X$ is a pair consisting of a category $S_X$ and a coverage? on $S_X$.
Then a morphism of sites $f : X \to Y$ is
a functor? $f^t : S_Y \to S_X$
such that the Yoneda extension? $\hat f^t : [S_Y^\op, Set] \to [S_X^{op}, Set]$ (of $Y_X \circ f^t : S_Y \to [S_X^{op}, Set]$) sends local isomorphism?s to local isomorphisms.
For $X$ a topological space, write $Sh(X) := Sh(Op(X))$ as usual for the topos given by the category of sheaves on the category of open subsets $Op(X)$ with the standard coverage
Given a morphism $f : X \to Y$ of site?s, the inverse image operation is a functor
that may be interpreted as encoding the idea of pulling back along $f$ the “bundle of wich the sheaf is the sheaf of sections”.
In the case that $X$ and $Y$ are (the site?s corresponding to) topological space?s this interpretation becomes literally true: the inverse image of a sheaf on topological spaces is the pullback operation on the corresponding etale space?s.
Consider a morphisms of sites? $f : X \to Y$ coming from a functor? $f^t : S_Y \to S_X$ of the underlying categories?.
The direct image? operation $f_* : PSh(X) \to PSh(Y)$ on presheaves? is just precomposition with $f^t$
The inverse image operation
on presheaves? is the left adjoint? to the direct image operation on presheaves, hence the left Kan extension?
of a presheaf? $F$ along $f^t$.
The inverse image operation on the category of sheaves? $Sh(Y) \subset PSh(Y)$ inside the category of presheaves involves Kan extension? followed by sheafification?.
First notice that
The direct image? operation $f_* : PSh(X) \to PSh(Y)$ restricts to a functor $f_* : Sh(X) \to Sh(Y)$ that sends sheaves to sheaves.
The direct image $f_* : PSh(X) \to PSh(Y)$ is more generally characterized by
where $\hat f^t$ is the Yoneda extension? of $Y \circ f^t : Y \to PSh(X)$ to a functor $\hat {f^t} : PSh(X) \to PSh(Y)$, because using the co-Yoneda lemma? and the colim expression for the Yoneda extension? we have
Let now $\pi : B \to A$ be a local isomorphism? in $PSh(Y)$. By definition of morphism of site?s we have that
is a local isomorphism? in $X$. From this and the above we obtain the isomorphism
where the isomorphism in the middle is due to the fact that $F$ is a sheaf on $X$. Since this holds for all local isomorphism $\pi : B \to A$ in $PSh(Y)$, $f_* F$ is a sheaf on $Y$.
For $f : X \to Y$ a morphism of site?s, the inverse image of sheaves is the functor
defined as the inverse image on presheaves followed by sheafification?
The inverse image $f^{-1} : Sh(Y) \to Sh(X)$ of sheaves has the following properties:
The left-adjointness is obtained by the following computation, for any two $F \in Sh(X)$ and $G \in Sh(Y)$ and using the above facts as well as the fact that sheafification? $\bar {(-)} : PSh(X) \to Sh(X)$ is left adjoint? to the inclusion $Sh(X) \hookrightarriw PSh(X)$:
The proof of left-exactness requires more technology and work.
In the case where the site?s $X$ and $Y$ in question are given by categories of open subsets? of topological space?s denoted, by a abuse of symbols, also by $X$ and $Y$, one can identify sheaves with their corresponding etale space?s over $X$ and $Y$. In that case the inverse image is simply obtained by the pullback along the continuous map $f : X \to Y$ of the corresponding etale space?s.
See also restriction and extension of sheaves?.
It follows that direct image and inverse image of sheaves define a geometric morphism? $f : Sh(X) \to Sh(Y)$ of sheaf? topoi?
Generally, therefore, the left adjoint partner in the adjoint pair defining a geometric morphism? of topoi? is called the inverse image functor.
The other adjoint to the direct image?, the right adjoint?, is (if it exists) the extension? of sheaves.
The standard example is that where $X$ and $Y$ are topological space?s and $S_X = Op(X)$, $S_Y = Op(Y)$ are their categories of open subsets.?
A continuous map $f : X \to Y$ induces the obvious functor $f^{-1} : Op(Y) \to Op(X)$, since preimages of open subsets under continuous maps are open.
Hence presheaves canonically push-forward
They do not in the same simple way pull back, since images of open subsets need not be open. The Kan extension computes the best possible approximation:
The inverse image $(f^{-1})^\dagger : PSh(Y) \to PSh(X)$ sends $F \in PSh(Y)$ to
This approximates the possibly non-open subset $f^{-1}(V)$ by all open subsets $U$ inside it.
On the other hand, the extension
$(f^{-1})^\ddagger : PSh(Y) \to PSh(X)$ sends $F \in PSh(Y)$ to
This approximates the possibly non-open subset $f^{-1}(V)$ by all open subsets $U$ containing it.
For every continuous map $f : X \to Y$ of Hausdorff topological spaces with the induced functor $f^{-1} : Op(Y) \to Op(X)$ of sites, the direct image
and the inverse image
constitute a geometric morphism
(denoted by the same symbol, by convenient abuse of notation).
This map $Hom_{Top}(X,Y) \to GeomMor(Sh(X),Sh(Y))$ is an bijection of sets.
That the induced pair $(f^*, f_*)$ forms a geometric morphism is (or should eventually be) discussed at inverse image.
We now show that the map is a bijection, i.e. that every geometric morphism of of sheaf topoi arises this way from a continuous function. We follow page 348 of
One reconstructs the continuous map $f : X \to Y$ from a geometric morphism $f : Sh(X) \to Sh(Y)$ as follows.
Write ${*} = Y \in Sh(Y)$ for the sheaf on $Op(Y)$ constant on the singleton set, the terminal object? in $Sh(Y)$.
Notice that since the inverse image $f^*$ preserves finite limits, every subobject $U_Y \hookrightarrow {*}$ is taken by $f^*$ to a subobject $U_X \hookrightarrow X$, obtained by applying $f^*$ to the pullback diagram
that characterizes the subobject $U_Y$ in the topos.
But, as the notation already suggests, the subobjects of $X,Y$ are just the open sets, i.e. the representable sheaves.
This yields a function $f^* : Obj(Op(Y)) \to Obj(Op(X))$ from open subsets to open subsets of which we know by assumption that it preserves finite limits and arbitrary colimits, i.e. finite intersections and arbitrary unions of open sets.
Using this define a function $\bar f : X \to Y$ of the sets underlying the topological spaces $X$ and $Y$ by setting
This yields a well defined function for the following reasons:
there is at most one $y$ satisfying this equation: if $y_1 \neq y_2$ both satisfy it, there are, by assumption of $Y$ being Hausdorff, neighbourhoods $V_1 \ni y_1$ and $V_2 \ni y_2$ such that (using that $f^*$ preserves limits hence intersections) $f^*(V_1) \cap f^*(V_2) = f^*(V_1 \cap V_2) = \emptyset$, which contradicts the assumption.
there is at least one $y$ satisfying this equation: again by contradiction: if there were none then every $y \in Y$ has a neighbourhood $V_y$ with $x \not\in f^*(V_y)$, so that similarly to above we conclude with $x \not\in \cup_{y \in Y} f^*(V_y) = f^*(\cup_y V_y) = f^*(Y) = X$ again a contradiction.
So our function $\bar f : X \to Y$ is well defined and satisfies $\bar f^{-1}(U_Y) = f^*(U_Y)$ for every open set $U_Y \in Obj(Op(Y))$. In particular it is therefore a continuous map.
It remains to check that this map reproduces the geometric morphism that we started with. For that we compute its direct image on any sheaf $A \in Sh(X)$ as
That is, it's a global element of the topos.
For the special case that $E = Sh(X)$ is the category of sheaves on a category of open subsets $Op(X)$ of a topological space $X$ this notion of point of a topos comes from the ordinary notion of points of $X$.
For notice that
$Set = Sh(*)$ is simply the topos of sheaves on a one-point space.
geometric morphisms $f : Sh(Y) \to Sh(X)$ between sheaf topoi are in bijection with continmuous functions of topological spaces $f : Y \to X$ (denoted by the same letter, by convenient abuse of notation).
It follows that for $E = Sh(X)$ points of $E$ in the sense of points of topoi are in bijection with the ordinary points of $X$.
The action of the direct image $x^* : Set \to Sh(X)$ and the inverse image $x_* : Sh(X) \to Set$ of a point $x : Set \to Sh(X)$ of a sheaf topos have special interpretation and relevance:
The direct image of a set $S$ under the point $x : {*} \to X$ is, by definition of direct image the sheaf
This is the skyscraper sheaf $skysc_x(S)$ with value $S$ supported at $X$. (In the first line on the right in the above we identify the set $S$ with the unique sheaf on the point it defines. Notice that $S(\emptyset) = pt$).
The inverse image of a sheaf $A$ under the point $x : {*} \to X$ is by definition of inverse image (see the Kan extension formula discussed there), the set
This is the stalk of $A$ at he point $x$,
By definition of geometric morphisms, taking the stalk at $x$ is left adjoint to forming the skyscraper sheaf at $x$:
for all $S \in Set$ and $A \in Sh(X)$ we have
The points $x \in X$ of the topological space $X$ are in canonical bijection with the points of $Sh(X)$ in the sense of point of a topos.
The stalk at $x$ of an object $e \in E$ is the image of $e$ under the corresponding inverse image? morphism
i.e.
For $E = Sh(X)$, the stalk $stalk_x(A)$ of a sheaf $A$ is given by the formula
By the general Kan extension formula for the inverse image (see there) one finds in this case for any sheaf $F \in Sh(X)$ the stalk
So for sheaves on (open subsets of) topological spaces the stalk at a given point is the colimit over all values of the sheaf on open subsets containing this point.
By the general definition of colimits in Set described at limits and colimits by example, the elements in this colimit can in turn be described as equivalence classes represented pairs $(z, V)$ with $x \in V$ $z \in F(V)$, where the equivalence relation says that two such pairs $(z_1, V_1)$ and $(z_2, V_2)$ coincide if there is a third pair $(z,U)$ with $U \subset V_1$ and $U \subset V_2$ such that $z = z_1|_U = z_2|_U$.
for $F = C(-)$ a sheaf of functions on $X$, such an equivalence class, hence such an element in a stalk of $F$ is called a function germ.
For $E = Sh(X)$ a topos of sheaves on a topological space (or generally of the topos $E$ has “enough points”), the behaviour of morphisms $f : A \to B$ in $E$ can be tested on stalks
A morphism $f : A \to B$ of sheaves on $X$ is a
resp. epimorphism
resp. isomorphism
if and only every induced map of stalk sets $stalk_x(f) : stalk_x(A) \to stalk_x(B)$ is, for all $x \in X$
The statement for isomorphisms follows from the identification of sheaves with etale spaces (e.g. section II, 6, corollary 3 in MacLane-Moerdijk, Sheaves in Geometry and Logic). The statement for epimorphisms/monomorphisms is proposition 6 there.
Let $X$ be a smooth manifold and let $\Omega^n(X)$ and $\Omega_{closed}^{n+1}(X)$ be the sheaves of differential $n$-forms and that of closed differential $(n+1)$-forms on $X$, respectively, for some $n \in \mathbb{N}$. Let
be the morphism of sheaves that is given on each open subset by the deRham differential.
Then:
for $U \subset X$ the map $d_U : \Omega^n(U) \to \Omega_{closed}^{n+1}(U)$ need not be epi, since not every closed form is exact;
but by the Poincaré lemma every closed form is locally exact, so that for each $x \in X$ the map of stalks $d_x : stalk_x(\Omega^n(X)) \to stalk_x(\Omega^{n+1}_{closed}(X))$ is an epimorphism.
Accordingly, the morphism $d : \Omega^n(X) \to \Omega^{n+1}_{closed}(X)$ is an epimorphism of sheaves.
This kind of example plays a crucial role in the computation of abelian sheaf cohomology, see the examples listed there.
Last revised on June 19, 2009 at 09:58:48. See the history of this page for a list of all contributions to it.