Shriek map

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In category theory, a branch of mathematics, certain unusual functors are denoted ${\displaystyle f_{!}}$ and ${\displaystyle f^{!},}$ 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:

• To distinguish a functor from a more usual functor ${\displaystyle f_{*}}$ or ${\displaystyle f^{*},}$ accordingly as it is covariant or contravariant.
• To indicate a map that goes "the wrong way" – a functor that has the same objects as a more familiar functor, but behaves differently on maps and has the opposite variance. For example, it has a pull-back where one expects a push-forward.

Examples

In algebraic geometry, these arise in image functors for sheaves, particularly Verdier duality, where ${\displaystyle f_{!}}$ is a "less usual" functor.

In

, or transfer maps. A fiber bundle ${\displaystyle F\to E\to B,}$ with base space B, fiber F, and total space E, has, like any other continuous map of topological spaces, a covariant map on homology ${\displaystyle H_{*}(E)\to H_{*}(B)}$ and a contravariant map on cohomology ${\displaystyle H^{*}(B)\to H^{*}(E).}$ However, it also has a covariant map on cohomology, corresponding in .

These can be used in understanding and proving the product property for the Euler characteristic of a fiber bundle.^[1]

Notes