Images of Sheaves
| Image functors for sheaves |
|---|
| direct image f∗ |
| inverse image f∗ |
| direct image with compact support f! |
| exceptional inverse image Rf! |
The definition of a morphism on sheaves makes sense only for sheaves on the same space X. This is because the data contained in a sheaf is indexed by the open sets of the space. If we have two sheaves on different spaces, then their data is indexed differently. There is no way to go directly from one set of data to the other.
However, it is possible to move a sheaf from one space to another using a continuous function. Let f : X → Y be a continuous function from a topological space X to a topological space Y. If we have a sheaf on X, we can move it to Y, and vice versa. There are four ways in which sheaves can be moved.
- A sheaf on X can be moved to Y using the direct image functor or the direct image with proper support functor .
- A sheaf on Y can be moved to X using the inverse image functor or the twisted inverse image functor .
The twisted inverse image functor is, in general, only defined as a functor between derived categories. These functors come in adjoint pairs: and are left and right adjoints of each other, and and are left and right adjoints of each other. The functors are intertwined with each other by Grothendieck duality and Verdier duality.
There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted and it is distinct from . See inverse image functor.
Read more about this topic: Sheaf (mathematics)
Famous quotes containing the words images of, images and/or sheaves:
“It was at a particular moment in the history of my own rages that I saw the Western world conditioned by the images of Marx, Darwin and Freud; and Marx, Darwin and Freud are the three most crashing bores of the Western world. The simplistic popularization of their ideas has thrust our world into a mental straitjacket from which we can only escape by the most anarchic violence.”
—William Golding (b. 1911)
“It was at a particular moment in the history of my own rages that I saw the Western world conditioned by the images of Marx, Darwin and Freud; and Marx, Darwin and Freud are the three most crashing bores of the Western world. The simplistic popularization of their ideas has thrust our world into a mental straitjacket from which we can only escape by the most anarchic violence.”
—William Golding (b. 1911)
“Being young you have not known
The fool’s triumph, nor yet
Love lost as soon as won,
Nor the best labourer dead
And all the sheaves to bind.”
—William Butler Yeats (1865–1939)