- Pullback (category theory)
In

category theory , a branch ofmathematics , a**pullback**(also called a**fibered product**or**Cartesian square**) is the limit of a diagram consisting of twomorphism s "f" : "X" → "Z" and "g" : "Y" → "Z" with a common codomain. The pullback is often written:$P\; =\; X\; imes\_Z\; Y.,$

**Universal property**Explicitly, the pullback of the morphisms "f" and "g" consists of an object "P" and two morphisms "p"

_{1}: "P" → "X" and "p"_{2}: "P" → "Y" for which the diagramcommutes. Moreover, the pullback ("P", "p"

_{1}, "p"_{2}) must be universal with respect to this diagram. That is, for any other such triple ("Q", "q"_{1}, "q"_{2}) there must exist a unique "u" : "Q" → "P" making the following diagram commute:As with all universal constructions, the pullback, if it exists, is unique up to a unique

isomorphism .**Weak pullbacks**A

**weak pullback**of a cospan "X" → "Z" ← "Y" is a cone over the cospan that is onlyweakly universal , that is, themediating morphism "u" : "Q" → "P" above need not be unique.**Examples**In the

category of sets the pullback of "f" and "g" is the set: $X\; imes\_Z\; Y\; =\; \{(x,\; y)\; in\; X\; imes\; Y|\; f(x)\; =\; g(y)\},,$

together with the restrictions of the

projection map s $pi\_1$ and $pi\_2$ to "X" ×_{"Z"}"Y" .*This example motivates another way of characterizing the pullback: as the equalizer of the morphisms "f" o "p"

_{1}, "g" o "p"_{2}: "X" × "Y" → "Z" where "X" × "Y" is the binary product of "X" and "Y" and "p"_{1}and "p"_{2}are the natural projections. This shows that pullbacks exist in any category with binary products and equalizers. In fact, by theexistence theorem for limits , all finite limits exist in a category with a terminal object, binary products and equalizers.Another example of a pullback comes from the theory of

fiber bundle s: given a bundle map π : "E" → "B" and acontinuous map "f" : "X" → "B", the pullback "X" ×_{"B"}"E" is a fiber bundle over "X" called thepullback bundle . The associated commutative diagram is a morphism of fiber bundles.In any category with a

terminal object "Z", the pullback "X" ×_{"Z"}"Y" is just the ordinary product "X" × "Y".**Properties***Whenever "X" ×

_{"Z"}"Y" exists, then so does "Y" ×_{"Z"}"X" and there is an isomorphism "X" ×_{"Z"}"Y" $cong$ "Y" ×_{"Z"}"X".

*Monomorphism s are stable under pullback: if the arrow "f" above is monic, then so is the arrow "p"_{2}. For example, in the category of sets, if "X" is a subset of "Z", then, for any "g" : "Y" → "Z", the pullback "X" ×_{"Z"}"Y" is theinverse image of "X" under "g".

*Isomorphism s are also stable, and hence, for example, "X" ×_{"X"}"Y" $cong$ "Y" for any map "Y" → "X".**See also*** The categorical dual of a pullback is a called a "pushout".

* Pullbacks in differential geometry

* Equijoin inrelational algebra .**References*** Adámek, Jií, Herrlich, Horst, & Strecker, George E.; (1990). [

*http://katmat.math.uni-bremen.de/acc/acc.pdf "Abstract and Concrete Categories"*] (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).

* Cohn, Paul M.; "Universal Algebra" (1981), D.Reidel Publishing, Holland, ISBN 90-277-1213-1 "(Originally published in 1965, by Harper & Row)".

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