Category-theoretic constructions for powerset construction in set theory

Question

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 ?

Answer 1

Here is an overview of how certain categorical constructions generalize set-theoretic ones:

• 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:

• 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 $A_i \subseteq A$)

Answer 2

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.