Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B. Care must be taken in the definition of Set to avoid set-theoretic paradoxes.

Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind.

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 ensure 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. Its zero morphisms are the empty functions ∅ → X.

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. One refers to collections that are not sets as proper classes. One can't handle proper classes as one handles sets; in particular, one can't write that those proper classes belong to a collection (either a set or a proper class). 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 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 and its powerset will belong to the universe). The existence of Grothendieck universes (other than the empty set and the set $V_{\omega }$ of all hereditarily finite sets) is not implied by the usual ZF axioms; it is an additional, independent axiom, roughly equivalent to the existence of strongly inaccessible cardinals. Assuming this extra axiom, one can limit the objects of Set to the elements of a particular universe. (There is no "set of all sets" within the model, but one can still reason about the class U of all inner sets, i. e., elements of U.)

In one variation of this scheme, the class of sets is the union of the entire tower of Grothendieck universes. (This is necessarily a proper class, but each Grothendieck universe is a set because it is an element of some larger Grothendieck universe.) However, one does not work directly with the "category of all sets". Instead, theorems are expressed in terms of the category Set_U whose objects are the elements of a sufficiently large Grothendieck universe U, and are then shown not to depend on the particular choice of U. As a foundation for category theory, this approach is well matched to a system like Tarski–Grothendieck set theory in which one cannot reason directly about proper classes; its principal disadvantage is that a theorem can be true of all Set_U but not of Set.

Various other solutions, and variations on the above, have been proposed.

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

See also

• Set theory
• Small set (category theory)

References

• Blass, A. The interaction between category theory and set theory. Contemporary Mathematics 30 (1984).
• Feferman, S. Set-theoretical foundations of category theory. Springer Lect. Notes Math. 106 (1969): 201–247.
• Lawvere, F.W. An elementary theory of the category of sets (long version) with commentary
• Mac Lane, S. One universe as a foundation for category theory. Springer Lect. Notes Math. 106 (1969): 192–200.
• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Springer. ISBN 0-387-98403-8. {{cite book}}: Unknown parameter |month= ignored (help) (Volume 5 in the series Graduate Texts in Mathematics)
• Pareigis, Bodo (1970), Categories and functors, Pure and applied mathematics, vol. 39, Academic Press, ISBN 978-0-12-545150-5