Shriek map

In category theory, a branch of mathematics, certain unusual functors are denoted formula_1 and formula_2 with the exclamation mark used to indicate that they are exceptional in some way. They are thus accordingly sometimes called shriek maps, with "shriek" being slang for an exclamation mark, though other terms are used, depending on context.
Usage.
Shriek notation is used in two senses:
Examples.
In algebraic geometry, these arise in image functors for sheaves, particularly
Verdier duality, where formula_1 is a "less usual" functor.
In algebraic topology, these arise particularly in fiber bundles, where they yield maps that have the opposite of the usual variance. They are thus called wrong way maps, Gysin maps, as they originated in the Gysin sequence, or transfer maps. A fiber bundle formula_6 with base space "B," fiber "F," and total space "E," has, like any other continuous map of topological spaces, a covariant map on homology formula_7 and a contravariant map on cohomology formula_8 However, it also has a covariant map on cohomology, corresponding in de Rham cohomology to "integration along the fiber", and a contravariant map on homology, corresponding in de Rham cohomology to "pointwise product with the fiber". The composition of the "wrong way" map with the usual map gives a map from the homology of the base to itself, analogous to a unit/counit of an adjunction; compare also Galois connection.
These can be used in understanding and proving the product property for the Euler characteristic of a fiber bundle.