nLab
inverse image

Idea

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

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

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 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

Given a morphisms of sites $f:X\to Y$ coming from a functor $f^t:S_Y\to S_X$ of the underlying categories.

on presheaves

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

$$\begin{array}{ccc}{S}_Y^{\mathrm{op}}& \stackrel{f_{*}F}{\to }& \mathrm{Set}\\ {\downarrow }^{f^t}& {\nearrow }_F\\ S_X^{\mathrm{op}}\end{array}.$$\array{ S_Y^{op} &\stackrel{f_* F}{\to}& Set \\ \downarrow^{f^t} & \nearrow_{F} \\ S_X^{op} } \,.

The inverse image operation

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

on presheaves is the left adjoint to the direct image operation on presheaves, hence the left Kan extension

$$f^{-1}F:={\mathrm{Lan}}_{f^t}F$$f^{-1} F := Lan_{f^t} F

of a presheaf $F$ along $f^t$.

on sheaves

The inverse image operation on the category of sheaves $\mathrm{Sh}(Y)\subset \mathrm{PSh}(Y)$ inside the category of presheaves involves Kan extension followed by sheafification.

First notice that

Lemma

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

Proof

The direct image $f_{*}:\mathrm{PSh}(X)\to \mathrm{PSh}(Y)$ is more generally characterized by

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

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

$$\begin{array}{rl}\mathrm{Hom}(A,f_{*}F)& \simeq \mathrm{Hom}({\mathrm{colim}}_{Y(U)\to A})U,f_{*}F)\\ & \simeq \underset{Y(U)\to A}{\lim }\mathrm{Hom}(U,f_{*}F)\\ & \simeq \underset{Y(U)\to A}{\lim }F(f^t(U))\\ & \simeq \mathrm{Hom}({\mathrm{colim}}_{Y(U)\to A}{f}^t(U),F)\\ & \simeq \mathrm{Hom}(\widehat{f^t}(A),F).\end{array}$$\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 $\pi :B\to A$ be a local isomorphism in $\mathrm{PSh}(Y)$. By definition of morphism of sites we have that

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

$$\mathrm{Hom}(B,f_{*}F)\simeq \mathrm{Hom}(\widehat{f^t}(B),F)\stackrel{\simeq }{\to }\mathrm{Hom}(\widehat{f^t}(A),F)\simeq \mathrm{Hom}(A,f_{*}F),$$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 $\mathrm{PSh}(Y)$, $f_{*}F$ is a sheaf on $Y$.

Definition

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

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

defined as the inverse image on presheaves followed by sheafification

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

Proposition

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

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

• 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 $F\in \mathrm{Sh}(X)$ and $G\in \mathrm{Sh}(Y)$ and using the above facts as well as the fact that sheafification $\overline{(-)}:\mathrm{PSh}(X)\to \mathrm{Sh}(X)$ is left adjoint to the inclusion $\mathrm{Sh}(X)hookrightarriw\mathrm{PSh}(X)$:

$$\begin{array}{rl}{\mathrm{Hom}}_{\mathrm{Sh}(Y)}(G,f_{*}F)& \simeq {\mathrm{Hom}}_{\mathrm{PSh}(Y)}(G,f_{*}F)\\ & \simeq {\mathrm{Hom}}_{\mathrm{PSh}(X)}({\mathrm{Lan}}_{f^t}G,F)\\ & \simeq {\mathrm{Hom}}_{\mathrm{Sh}(X)}(\overline{({\mathrm{Lan}}_{f^t}G)},F)\\ & =:{\mathrm{Hom}}_{\mathrm{Sh}(X)}(f^{-1}G,F)\end{array}.$$\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 a 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.

Remarks

• See also restriction and extension of sheaves.

• It follows that direct image and inverse image of sheaves define a geometric morphism $f:\mathrm{Sh}(X)\to \mathrm{Sh}(Y)$ 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.

Examples

The standard example is that where $X$ and $Y$ are topological spaces and $S_X=\mathrm{Op}(X)$, $S_Y=\mathrm{Op}(Y)$ are their categories of open subsets.

A continuous map $f:X\to Y$ induces the obvious functor $f^{-1}:\mathrm{Op}(Y)\to \mathrm{Op}(X)$, since preimages of open subsets under continuous maps are open.

Hence presheaves canonically push-forward

$$f_{*}^{-1}:\mathrm{PSh}(X)\to \mathrm{PSh}(Y)$$f^{-1}_* : 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 $(f^{-1}{)}^{†}:\mathrm{PSh}(Y)\to \mathrm{PSh}(X)$ sends $F\in \mathrm{PSh}(Y)$ to

$$f^{†}F:U\mapsto {\mathrm{colim}}_{(U\to f^{-1}(V))\in ({\mathrm{const}}_U,f^{-1})}F(V).$$f^\dagger F : U \mapsto colim_{(U \to f^{-1}(V)) \in (const_U, f^{-1})} F(V) \,.

This approximates the possibly non-open subset $f^{-1}(V)$ by all open subsets $U$ inside it.

On the other hand, the extension

$(f^{-1}{)}^{‡}:\mathrm{PSh}(Y)\to \mathrm{PSh}(X)$ sends $F\in \mathrm{PSh}(Y)$ to

$$f^{†}F:U\mapsto {\mathrm{colim}}_{(f^{-1}(V)\to U)\in (f^{-1},{\mathrm{const}}_U)}F(V).$$f^\dagger F : U \mapsto colim_{(f^{-1}(V) \to U) \in (f^{-1},const_U)} F(V) \,.

This approximates the possibly non-open subset $f^{-1}(V)$ by all open subsets $U$ containing it.

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 etale spaces see secton VII.1 of

• MacLane-Moerdijk, Sheaves in Geometry and Logic