List of mathematical abbreviations

2013-11-25T19:22:28Z

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In [[category theory]], a branch of [[mathematics]], certain unusual [[functor]]s are denoted <math>f_!</math> and <math>f^!,</math> 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 <math>f_*</math> or <math>f^*,</math> 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 <math>f_!</math> is a "less usual" functor.

In [[algebraic topology]], these arise particularly in [[fiber bundle]]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]], or '''transfer maps.''' A fiber bundle <math>F \to E \to B,</math> with base space ''B,'' fiber ''F,'' and total space ''E,'' has, like any other continuous map of topological spaces, a covariant map on homology <math>H_*(E) \to H_*(B)</math> and a contravariant map on cohomology <math>H^*(B) \to H^*(E).</math> 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#Fibration property|Euler characteristic of a fiber bundle]].<ref>{{citation
|title=Fibre bundles and the Euler characteristic
|first=Daniel Henry
|last=Gottlieb
|journal=Journal of Differential Geometry
|volume=10
|issue=1
|year=1975
|pages=39–48
|url=http://www.math.purdue.edu/~gottlieb/Bibliography/17FibreBundlesAndtheEulerCharacteristic.pdf
}}</ref>

== Notes ==
{{reflist}}

[[Category:Mathematical notation]]
[[Category:Algebraic geometry]]
[[Category:Algebraic topology]]

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List of mathematical abbreviations