nLab inverse image

This page is about inverse images of sheaves and related subjects. For the set-theoretic operation, see preimage.

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Idea

An inverse image operation is the left adjoint part $f^*$ of a geometric morphism $(f^* \dashv f_*) \colon E \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} F$ between topoi.

For the case of sheaf topoi the inverse image

$$f^{-1} : Sh(Y) \to Sh(X)$$

may be interpreted as encoding the idea of pulling back along $f$ the “bundle of which the sheaf is the sheaf of sections”.

In the case that $X$ and $Y$ are (the sites corresponding to) topological spaces, this interpretation becomes literally true: the inverse image of a sheaf on topological spaces is the pullback operation on the corresponding etale spaces.

Definition

Consider a functor $f^t \colon Y \to X$ between the underlying categories of sites.

on presheaves

The direct image operation $f_* \colon PSh(X) \to PSh(Y)$ on presheaves is just precomposition with $f^t$

$$\array{ Y^{op} &\overset{f_* F}{\longrightarrow}& Set \\ \mathllap{ {}^{f^t} }\big\downarrow & \nearrow\mathrlap{_{F}} \\ X^{op} } \,.$$

The inverse image operation

$$f^{-1} \colon PSh(Y) \to PSh(X)$$

on presheaves is the left adjoint to the direct image operation on presheaves, hence the left Kan extension along $f^t$:

$$f^{-1} F \;\coloneqq\; Lan_{f^t} F \,.$$

.

on sheaves

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

Lemma

The direct image operation $f_* \colon PSh(X) \to PSh(Y)$ restricts to a functor $f_* \colon Sh(X) \to Sh(Y)$ that sends sheaves to sheaves.

Proof

The direct image $f_* \colon PSh(X) \to PSh(Y)$ is more generally characterized by

$$Hom_{PSh(Y)}(A, f_* F) \simeq Hom_{PSh(X)}(\hat {f^t} A, F)$$

where $\hat f^t$ is the Yoneda extension of $Y \circ f^t \colon Y \to PSh(X)$ to a functor $\hat {f^t} : PSh(Y) \to PSh(X)$, because using the co-Yoneda lemma and the colim expression for the Yoneda extension we have

$$\begin{aligned} Hom(A, f_* F) & \simeq Hom(colim_{Y(U) \to A}) U, f_* F) \\ & \simeq \lim_{Y(U) \to A} Hom(U, f_* F) \\ & \simeq \lim_{Y(U) \to A} F(f^t(U)) \\ & \simeq Hom( colim_{Y(U) \to A} f^t(U), F ) \\ & \simeq Hom(\hat {f^t}(A), F) \,. \end{aligned}$$

Let now $\pi \colon B \to A$ be a local isomorphism in $PSh(Y)$. By definition of morphism of sites we have that

$$\hat {f^t}(\pi) : \hat{f^t}(B) \to \hat{f^t}(A)$$

is a local isomorphism in $X$. From this and the above we obtain the isomorphism

$$Hom(B, f_* F) \simeq Hom(\hat {f^t}(B), F) \stackrel{\simeq}{\to} Hom(\hat {f^t}(A), F) \simeq Hom(A, f_* F) \,,$$

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 $Y$.

Definition

For $f : X \to Y$ a morphism of sites, the inverse image of sheaves is the functor

$$f^{-1} : Sh(Y) \to Sh(X)$$

defined as the inverse image on presheaves followed by sheafification

$$f^{-1} : Sh(Y) \hookrightarrow PSh(Y) \stackrel{Lan_{f^t}}{\to} PSh(X) \stackrel{\bar{-}}{\to} Sh(X) \,.$$

Proposition

The inverse image $f^{-1} : Sh(Y) \to Sh(X)$ of sheaves has the following properties:

• it is left adjoint to the direct image $(f^{-1} \dashv f_*)$;

• it therefore commutes with small colimits but is in addition left exact in that it commutes with finite limits.

Proof

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 $Sh(X) \hookrightarrow PSh(X)$:

$$\begin{aligned} Hom_{Sh(Y)}(G, f_*F) & \simeq Hom_{PSh(Y)}(G, f_* F) \\ & \simeq Hom_{PSh(X)}(Lan_{f^t} G, F) \\ & \simeq Hom_{Sh(X)}( \bar{(Lan_{f^t} G)}, F) \\ & =: Hom_{Sh(X)}(f^{-1}G, F) \end{aligned} \,.$$

The proof of left-exactness requires more technology and work.

on sheaves on topological spaces

In the case where the sites $X$ and $Y$ in question are given by categories of open subsets of topological spaces denoted, by an abuse of symbols, also by $X$ and $Y$, one can identify sheaves with their corresponding etale spaces 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 spaces.

Properties

• 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 (or abelian categories of quasicoherent sheaves) is called the inverse image functor. In fact more general in geometry, including noncommutative morphisms often induce or are defined via pairs of adjoint functors among some associated categories of objects over a geometric space; then the left adjoint part is called the inverse image part. Geometers also often say inverse image for an arbitrary functor of the form $f^*$ in a fibered category. For abelian categories of sheaf-like objects, the corresponding higher derived functors of inverse image functors are sometimes called higher (derived) inverse image functors.

• The other adjoint to the direct image, the right adjoint, is (if it exists) the extension of sheaves.

Examples

The standard example is that where $X$ and $Y$ are topological spaces 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

$$f_* : PSh(X) \to PSh(Y)$$

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

$$f^\dagger F(U) = \colim_{U \subseteq f^{-1}(V)} F(V) \,.$$

What’s this really doing is using $V$ to approximate $f(U)$ from the outside. Thus a section of $f^\dagger F$ on $U$ will be an equivalence class of sections of $F$ on open neighbourhoods of $f(U)$, under the equivalence given by agreement on a restriction to a smaller neighbourhood:

$$(s, V_1) \sim (t, V_2) \quad\underline{iff}\quad \exists W \in Op(Y), V_1 \cap V_2 \supseteq W \supseteq f(U), \; s\vert_W = t\vert_W.$$

Compare this with the definition of germs at a stalk.

On the other hand, the extension $(f^{-1})^\ddagger : PSh(Y) \to PSh(X)$ sends $F \in PSh(Y)$ to

$$f^\ddagger F(U) = \lim_{f^{-1}(V) \subseteq U} F(V) \,.$$

This approximates the possibly non-open subset $f(U)$ by all open subsets $V$ contained in it. This corresponds to taking the right Kan extension instead of the left one, and when it exists it’s called extension of presheaves.

Related concepts

• derived inverse image

• exceptional inverse image

References

For the general description in terms of Kan extension and sheafification see section 17.5 of

• Kashiwara-Schapira, Categories and Sheaves

For the description in terms of pullback of étale spaces, see section VII.1 of

• MacLane-Moerdijk, Sheaves in Geometry and Logic