In category theory, a branch of mathematics, certain unusual

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

s are denoted

f_!

and

f^!,

with the

exclamation mark The exclamation mark, , or exclamation point (American English), is a punctuation mark usually used after an interjection or exclamation to indicate strong feelings or to show emphasis. The exclamation mark often marks the end of a sentence, ...

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_*

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 In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...

where one expects a push-forward.

Examples

In algebraic geometry, these arise in

image functors for sheaves In mathematics, especially in sheaf theory—a domain applied in areas such as topology, logic and algebraic geometry—there are four image functors for sheaves that belong together in various senses. Given a continuous mapping ''f'': ''X'' � ...

, particularly

Verdier duality In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of Alexander Groth ...

, where

f_!

is a "less usual" functor. In

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...

, these arise particularly in

fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...

s, 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 In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool fo ...

, or transfer maps. A fiber bundle

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

H_*(E) \to H_*(B)

and a contravariant map on cohomology

H^*(B) \to H^*(E).

However, it also has a covariant map on cohomology, corresponding in

de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...

to "

integration along the fiber In differential geometry, the integration along fibers of a ''k''-form yields a (k-m)-form where ''m'' is the dimension of the fiber, via "integration". It is also called the fiber integration. Definition Let \pi: E \to B be a fiber bundle over ...

", 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 In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...

of an adjunction; compare also

Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fu ...

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

Notes

{{reflist Mathematical notation Algebraic geometry Algebraic topology