Category-theoretic constructions for powerset construction in set theory Set theory has fundamental constructions such as the powerset. What is universal construction in the category-theoretic foundation for this construction ? Any reference for other set fundamental constructions ?
 A: Here is an overview of how certain categorical constructions generalize set-theoretic ones:


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*cartesian products $\prod_{i\in I} A_i$ are generalized by products

*disjoint unions $\sum_{i\in I} A_i$ are generalized by coproducts

*sets $B^A$ of functions $A \to B$ is generalized by exponentials

*sets of equivalence classes are generalized by coequalizers of (internal) congruences

*the empty set is generalized by initial objects

*singletons are generalized by terminal objects

*two-element sets may be generalized by $1+1$, where $1$ is a terminal object and $+$ is a coproduct; or by subobject classifiers


The powerset can be generalized as an exponential $X^A$ where $X = 1 + 1$ or $X$ is a subobject classifier. This would be an internal generalization.
On the other hand, a powerset can also be generalized externally by the preset/poset $\operatorname{Sub} A$ of subobjects of an object $A$. Then:


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*unions $\bigcup_{i\in I} A_i$ are generalized by suprema in $\operatorname{Sub} A$ (if $A_i \subseteq A$)

*intersections $\bigcap_{i\in I} A_i$ are generalized by infima in $\operatorname{Sub} A$ (if $A_i \subseteq A$)

A: In addition to the answer from Stefan, the notion of power object generalizes the notion of power set from the category Set to an arbitrary category with finite limits. If $1$ is a terminal object, then the power object is precisely a subobject classifier. A power object in Set is precisely a power set. A category with finite limits and power objects for all objects is precisely a topos. The power object $PA$ for any object $A$ in the typos is the exponential object into the subject classifier.
