Shriek map

In category theory, a branch of mathematics, certain unusual functors are denoted

f

!

{\displaystyle f_{!}}

and

f

!

,

{\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

f

∗

{\displaystyle f_{*}}

or

f

∗

,

{\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

f

!

{\displaystyle f_{!}}

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.