Definition Suppose X,Y,Z are sets, and we have maps

f:Y Z,
Φ:X Y.

Then the pullback of f under Φ is the mapping

Φf:X Z,
x (fΦ)(x).

Let us denote by M(X,Y) the set of all mappings f:XY. We then see that Φ is a mapping M(Y,Z)M(X,Z). In other words, Φ pulls back the set where f is defined on from Y to X. This is illustrated in the below diagram.


0.0.1 Properties

  1. 1.

    For any set X, (idX)=idM(X,X).

  2. 2.

    Suppose we have maps

    Φ:X Y,
    Ψ:Y Z

    between sets X,Y,Z. Then

  3. 3.

    If Φ:XY is a bijection, then Φ is a bijection and

  4. 4.

    Suppose X,Y are sets with XY. Then we have the inclusion mapMathworldPlanetmath ι:XY, and for any f:YZ, we have


    where f|X is the restrictionPlanetmathPlanetmathPlanetmath (http://planetmath.org/RestrictionOfAFunction) of f to X.

Title pullback
Canonical name Pullback
Date of creation 2013-03-22 13:50:04
Last modified on 2013-03-22 13:50:04
Owner matte (1858)
Last modified by matte (1858)
Numerical id 14
Author matte (1858)
Entry type Definition
Classification msc 03-00
Related topic InclusionMapping
Related topic RestrictionOfAFunction
Related topic PullbackOfAKForm