- 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:
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 diagram
commutes. 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 only
weakly 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:
together with the restrictions of the
projection map s and 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 the
existence 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" "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" "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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