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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 ?

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    $\begingroup$ That is a particular case of an exponential object. $\endgroup$ – Stefan Perko Sep 22 '16 at 19:37
  • $\begingroup$ Could you also name some other set constructions ? Are they all have associated constructions ? $\endgroup$ – Tom Sep 22 '16 at 19:43
  • $\begingroup$ @StefanPerko This was my first thought after googling 'power set universal property' but we have to specify the $2$ object which I guess is the binary direct sum of the terminal object if we were to try to generalize this beyond $\bf{Set}$? Thoughts? $\endgroup$ – basket Sep 22 '16 at 19:45
  • $\begingroup$ @basket: may some subsets be intersected, correct ? $\endgroup$ – Tom Sep 22 '16 at 19:54
  • $\begingroup$ @basket $1+1 = 2$. (no joke intended) $\endgroup$ – Stefan Perko Sep 22 '16 at 20:04
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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 $A_i \subseteq A$)
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  • $\begingroup$ $\Omega = 2$ only happens in special cases. $\endgroup$ – user14972 Sep 22 '16 at 20:22
  • $\begingroup$ @Hurkyl Right, I should not have called it "$2$". $\endgroup$ – Stefan Perko Sep 22 '16 at 20:22
  • $\begingroup$ @StefanPerko, Can you add generalization of limit/colimit, representable functor, adjoint functor, end/coend, Kan extension, dependent sum/dependent product ? $\endgroup$ – Tom Sep 22 '16 at 20:31
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    $\begingroup$ @Tom ... uhm. All of these are adjoints, all of these are representables, all of these are Kan extensions and all of these are limits / colimits. And they are countless ways to generalize all of these too. That's a bit to broad to answer and in any case should be a seperate question, I think. $\endgroup$ – Stefan Perko Sep 22 '16 at 20:35
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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.

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