inverse image

Inverse images


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

Inverse images


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

Given a morphism f:XYf : X \to Y of sites, the inverse image operation of the induced geometric morphism Sh(X)Sh(Y)Sh(X) \to Sh(Y) on categories of sheaves is a functor

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

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

In the case that XX and YY 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.


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

on presheaves

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

S Y op f *F Set f t F S X op. \array{ S_Y^{op} &\stackrel{f_* F}{\to}& Set \\ \downarrow^{f^t} & \nearrow_{F} \\ S_X^{op} } \,.

The inverse image operation

f 1:PSh(Y)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 1F:=Lan f tF f^{-1} F := Lan_{f^t} F

of a presheaf FF along f tf^t.

on sheaves

The inverse image operation on the category of sheaves Sh(Y)PSh(Y)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)PSh(Y)f_* : PSh(X) \to PSh(Y) restricts to a functor f *:Sh(X)Sh(Y)f_* : Sh(X) \to Sh(Y) that sends sheaves to sheaves.


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

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

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

Hom(A,f *F) Hom(colim Y(U)A)U,f *F) lim Y(U)AHom(U,f *F) lim Y(U)AF(f t(U)) Hom(colim Y(U)Af t(U),F) Hom(f t^(A),F). \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 π:BA\pi : B \to A be a local isomorphism in PSh(Y)PSh(Y). By definition of morphism of sites we have that

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

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

Hom(B,f *F)Hom(f t^(B),F)Hom(f t^(A),F)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 FF is a sheaf on XX. Since this holds for all local isomorphism π:BA\pi : B \to A in PSh(Y)PSh(Y), f *Ff_* F is a sheaf on YY.


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

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

defined as the inverse image on presheaves followed by sheafification

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

The inverse image f 1:Sh(Y)Sh(X)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 FSh(X)F \in Sh(X) and GSh(Y)G \in Sh(Y) and using the above facts as well as the fact that sheafification ()¯:PSh(X)Sh(X)\bar {(-)} : PSh(X) \to Sh(X) is left adjoint to the inclusion Sh(X)PSh(X)Sh(X) \hookrightarrow PSh(X):

Hom Sh(Y)(G,f *F) Hom PSh(Y)(G,f *F) Hom PSh(X)(Lan f tG,F) Hom Sh(X)((Lan f tG)¯,F) =:Hom Sh(X)(f 1G,F). \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 XX and YY in question are given by categories of open subsets of topological spaces denoted, by an abuse of symbols, also by XX and YY, one can identify sheaves with their corresponding etale spaces over XX and YY. In that case the inverse image is simply obtained by the pullback along the continuous map f:XYf : X \to Y of the corresponding etale spaces.


  • See also restriction and extension of sheaves.

  • It follows that direct image and inverse image of sheaves define a geometric morphism f:Sh(X)Sh(Y)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 *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.


The standard example is that where XX and YY are topological spaces and S X=Op(X)S_X = Op(X), S Y=Op(Y)S_Y = Op(Y) are their categories of open subsets.

A continuous map f:XYf : X \to Y induces the obvious functor f 1:Op(Y)Op(X)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)PSh(Y) 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) :PSh(Y)PSh(X)(f^{-1})^\dagger : PSh(Y) \to PSh(X) sends FPSh(Y)F \in PSh(Y) to

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

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

(s,V 1)(t,V 2)iff̲WOp(Y),V 1V 2Wf(U),s| W=t| W. (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) :PSh(Y)PSh(X)(f^{-1})^\ddagger : PSh(Y) \to PSh(X) sends FPSh(Y)F \in PSh(Y) to

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

This approximates the possibly non-open subset f(U)f(U) by all open subsets VV 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.


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

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

Last revised on January 21, 2021 at 03:39:37. See the history of this page for a list of all contributions to it.