Properties
- Whenever X ×Z Y exists, then so does Y ×Z X and there is an isomorphism X ×Z Y Y ×Z X.
- Monomorphisms are stable under pullback: if the arrow f above is monic, then so is the arrow p2. 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 the inverse image of X under g.
- Isomorphisms are also stable, and hence, for example, X ×X Y Y for any map Y → X.
- Any category with pullbacks and products has equalizers.
Read more about this topic: Pullback (category Theory)
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)