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Cartesian square (category theory) : ウィキペディア英語版
Pullback (category theory)
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 and with a common codomain; it is the limit of the cospan . The pullback is often written
:.
The categorical dual of a pullback is a called a ''pushout''. Remarks opposite to the above apply: the pushout is a coproduct with additional structure.
==Universal property==
Explicitly, the pullback of the morphisms and consists of an object and two morphisms and for which the diagram
:
commutes. Moreover, the pullback must be universal with respect to this diagram. That is, for any other such triple for which the following diagram commutes, there must exist a unique (called a mediating morphism) such that
:p_2 \circ u=q_2, \qquad p_1\circ u=q_1.
:
As with all universal constructions, the pullback, if it exists, is unique up to isomorphism. In fact, given two pullbacks and of the same cospan, there is a unique isomorphism between and respecting the pullback structure.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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