About: Pullback (category theory)

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In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is often written P = X ×Z Y The dual concept of the pullback is the pushout.

Attributes	Values
rdf:type	stream
rdfs:label	Faserprodukt (de) Produit fibré (fr) 당김 (범주론) (ko) 引き戻し (圏論) (ja) Pullback (category theory) (en) Produkt włóknisty (pl) Расслоённое произведение (ru) Produto fibrado (pt) 拉回 (范畴论) (zh) Розшарований добуток (uk)
rdfs:comment	Das Faserprodukt (auch Pullback, kartesisches Quadrat oder Pullback-Quadrat) ist ein Begriff aus dem mathematischen Teilgebiet der Kategorientheorie. Zentrale Bedeutung kommt dem Faserprodukt in der algebraischen Geometrie zu. Der Begriff des Faserproduktes ist dual zum Begriff des Pushout. (de) 범주론에서 당김(영어: pullback 풀백[*])은 어떤 한 쌍의 사상에 의해 결정되는, 곱의 일반화이다. 일부 범주에서는 흔히 올곱(미국 영어: fibered product, 영국 영어: fibred product)이라고 불린다. (ko) 圏論という数学の分野において，引き戻し（ひきもどし，英: pullback），あるいはファイバー積 (fiber/fibre/fibered product)，デカルトの四角形 (Cartesian square) とは，共通の終域を持つ2つの射 f: X → Z, g: Y → Z からなる図式の極限である．引き戻しはしばしば P = X ×Z Y と書かれ，2つの自然な射 P → X, P → Y を備えている．2つの射の引き戻しが存在するとは限らないが，存在すれば2つの射から本質的に一意に定義される．多くの状況において，X ×Z Y は，元 x ∈ X と y ∈ Y の対 (x, y) であって f(x) = g(y) なるものからなるものと直観的に考えることができる．一般の定義には普遍性が用いられ，このことを本質的な理由として，引き戻しは2つの与えられた射を可換四角形に適合させる「最も一般の」方法である．引き戻しの双対概念は (pushout) である． (ja) O produto fibrado (ou pullback) é uma construção de teoria das categorias. (pt) Расслоённое произведение (послойное произведение, коамальгама, декартов квадрат, англ. pullback) — теоретико-категорное понятие, определяемое как предел диаграммы, состоящей из двух морфизмов: Расслоённое произведение часто обозначают как Двойственное понятие — кодекартов квадрат. (ru) Розшарований добуток (також декартів квадрат) — теоретико-категорне поняття, яке можна задати як границю , що складається з двох морфізмів: Розшарований добуток позначається Двоїстим поняттям є розшарований кодобуток. (uk) 在数学分支范畴论中，拉回（也称为纤维积或笛卡尔方块）是由具有公共上域的两个态射f : X → Z与g : Y → Z组成的的极限。拉回经常写作 (zh) In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is often written P = X ×Z Y The dual concept of the pullback is the pushout. (en) En mathématiques, le produit fibré est une opération entre deux ensembles munis tous deux d'une application vers un même troisième ensemble. Sa définition s'étend à certaines catégories en satisfaisant une propriété universelle de factorisation de diagrammes, en dualité avec la somme amalgamée. (fr)
foaf:depiction
dcterms:subject	Limits (category theory)
Wikipage page ID	841689 (xsd:integer)
Wikipage revision ID	1118862053 (xsd:integer)
Link from a Wikipage to another Wikipage	Product ring Monomorphism Paul Cohn Relational algebra Inclusion map Continuous map Mathematics Subring Pullback (differential geometry) Pullback bundle Monoid Morphism Cone (category theory) Equaliser (mathematics) Limit (category theory) Commutative diagram Commutative ring Horst Herrlich Span (category theory) Tangent space Category of sets Dual (category theory) Cospan Fiber bundle Diagram (category theory) Discrete category Isomorphism Product (category theory) Restriction (mathematics) Terminal object Abelian category Kernel (category theory) Least common multiple Limits (category theory) Smoothness Disjoint union Fiber product of schemes Continuous function (topology) Existence theorem for limits Integer Category of groups Category of modules Category of topological spaces Category theory Exact sequence Pushout (category theory) Universal property Injective Subobject Ring homomorphism Preimage Category of commutative rings Projection map Object (category theory) dbr:Weakly_universal_property
Link from a Wikipage to an external page	http://katmat.math.uni-bremen.de/acc/acc.pdf https://web.archive.org/web/20080916162345/http:/www.j-paine.org/cgi-bin/webcats/webcats.php
sameAs	Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory) Pullback (category theory)

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