Pullback (category Theory) - Examples

Examples

In the category of sets, a pullback of f and g is given by the set

together with the restrictions of the projection maps and to X ×Z Y .

Alternatively one may view the pullback in Set asymmetrically:

\cong \coprod_{x\in X} g^{-1}
\cong \coprod_{y\in Y} f^{-1}

where is the disjoint (tagged) union of sets (the involved sets are not disjoint on their own unless f resp. g is injective). In the first case, the projection extracts the x index while forgets the index, leaving elements of Y.

  • This example motivates another way of characterizing the pullback: as the equalizer of the morphisms f p1, g p2 : X × Y → Z where X × Y is the binary product of X and Y and p1 and p2 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 bundles: given a bundle map π : E → B and a continuous map f : X → B, the pullback X ×B E is a fiber bundle over X called the pullback 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.

Read more about this topic:  Pullback (category Theory)

Famous quotes containing the word examples:

    “There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
    —Bernard Mandeville (1670–1733)

    “No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
    —André Breton (1896–1966)

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)