Category of sets

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In mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics.

The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.

The empty set serves as the initial object in Set with empty functions as morphisms. Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no zero objects in Set.

The category Set is complete and co-complete. The product in this category is given by the cartesian product of sets. The coproduct is given by the disjoint union: given sets Ai where i ranges over some index set I, we construct the coproduct as the union of Ai×{i} (the cartesian product with i serves to insure that all the components stay disjoint).

Set is the prototype of a concrete category; other categories are concrete if they "resemble" Set in some well-defined way.

Every two-element set serves as a subobject classifier in Set. The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B. Set is thus a topos (and in particular cartesian closed).

Set is not abelian, additive or preadditive; it does not even have zero morphisms.

Every not initial object in Set is injective and (assuming the axiom of choice) also projective.

[edit] The size of the category of sets

It is often assumed that the collection of all sets is not a set. For instance, this follows from the axiom of foundation. This leads to problems when formalizing what the category Set really is.

One way to resolve this difficulty is to say that the collection of all sets is a proper class. One can then work in a framework, such as NBG set theory, that distinguishes between sets and classes. The category Set is then said to be large.

Another solution is to assume the existence of Grothendieck universes. The objects of Set are then sets relative to some universe, U. The collection of all these sets is again a set, but not a set in U.

Various other solutions, and variations on the above, have been proposed[1][2][3].

[edit] References

  1. ^ Mac Lane, S. One universe as a foundation for category theory. Springer Lect. Notes Math. 106 (1969): 192–200.
  2. ^ Feferman, S. Set-theoretical foundations of category theory. Springer Lect. Notes Math. 106 (1969): 201–247.
  3. ^ Blass, A. The interaction between category theory and set theory. Contemporary Mathematics 30 (1984).
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