Category of sets

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.

Properties of the category of sets

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 "A"_"i" where "i" ranges over some index set "I", we construct the coproduct as the union of "A"_"i"×{"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.

Foundations for the category of sets

In Zermelo–Fraenkel set theory the collection of all sets is not a set, this follows from the axiom of foundation. In this situation, one refers to the collection of all sets as a proper class. In ZF set theory, proper classes do not have any formal status. This is a problem: it means that the category of sets cannot be formalized straightforwardly in this setting.

One way to resolve the problem is to work in a system that gives formal status to proper classes, such as NBG set theory. In this setting, categories formed from sets are said to be "small" and those (like Set) that are formed from proper classes are said to be be "large".

Another solution is to assume the existence of Grothendieck universes. Roughly speaking, a Grothendieck universe is a set which is itself a model of ZF(C) (for instance if a set belongs to a universe, its elements or its powerset will belong to the universe). The existence of universes is not implied by the usual ZF axioms; it is an additional, independent axiom. Assuming this extra axiom, the objects of Set can be defined to be the elements of a particular universe, rather than all the sets.

Various other solutions, and variations on the above, have been proposed [Mac Lane, S. One universe as a foundation for category theory. Springer Lect. Notes Math. 106 (1969): 192–200. ] [Feferman, S. Set-theoretical foundations of category theory. Springer Lect. Notes Math. 106 (1969): 201–247.] [Blass, A. [http://www.math.lsa.umich.edu/~ablass/interact.pdf The interaction between category theory and set theory] . Contemporary Mathematics 30 (1984).] .

The same issues arise with other concrete categories, such as the category of groups or the category of topological spaces.

References

* (Volume 5 in the series Graduate Texts in Mathematics)