In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map
, the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X.
The direct image functor is the primary operation on sheaves, with the simplest definition.
The inverse image exhibits some relatively subtle features.
Suppose we are given a sheaf
and that we want to transport
using a continuous map
We will call the result the inverse image or pullback sheaf
If we try to imitate the direct image by setting for each open set
, we immediately run into a problem:
is not necessarily open.
The best we could do is to approximate it by open sets, and even then we will get a presheaf and not a sheaf.
to be the sheaf associated to the presheaf: (Here
is an open subset of
and the colimit runs over all open subsets
is just the inclusion of a point
is just the stalk of
at this point.
The restriction maps, as well as the functoriality of the inverse image follows from the universal property of direct limits.
When dealing with morphisms
of locally ringed spaces, for example schemes in algebraic geometry, one often works with sheaves of
is the structure sheaf of
is inappropriate, because in general it does not even give sheaves of
In order to remedy this, one defines in this situation for a sheaf of
its inverse image by However, the morphisms
denotes the inclusion of a closed subset, the stalk of
is canonically isomorphic to
A similar adjunction holds for the case of sheaves of modules, replacing