The notion of power object generalizes the notion of power set from the category Set to an arbitrary category with finite limits.
Let be a category with finite limits. A power object of an object is
an object
such that
If may lack some finite limits, then we may weaken that condition as follows:
If has all pullbacks (but may lack products), then equip each of and with a jointly monic pair of morphisms, one to and one to or , in place of the single monomorphism to the product of these targets; must then be the joint pullback
If may lack some pullbacks, then we simply require that the pullback that is to equal must exist. But arguably we should require, if is to be a power object, that this pullback exists for any given map .
If is a terminal object, then is precisely a subobject classifier.
A category with finite limits and power objects for all objects is precisely a topos. The power object of any object in the topos is the exponential object into the subobject classifier.
See Trimble on ETCS I for the axiom of power sets in the elementary theory of the category of sets.
Last revised on November 13, 2022 at 14:06:19. See the history of this page for a list of all contributions to it.