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.

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.

Trouble with the category of sets

In ZFC the collection of all sets ( denoted $V$ ) is not a set (this is usually proved using the axiom of foundation), it is a proper classe. This has bad consequences since we can't say that $V$ belongs to something (in ZFC $A\in B$ has the consequence that $A$ is a set). Thus we run into trouble in formalizing category Set the straightforward way since a category is a n-uple and one of the element of this n-uple should be the collection of all sets.

One way to resolve the problem is to give up working in ZFC and choose a more amiable framework. For instance the NBG set theory. In this theory the properties of proper classes are less retrictive. In this setting categories which are sets are said to be small and those (like Set) which are proper classes are said to be be large.

If we want to continue to use ZFC we can use the concept of Grothendieck universe. A universe is a set which behaves such as we expect $V$ behaves (for instance if a set belongs to a universe, its elements or its powerset will belong to the universe). Working with universes however supposes we add an extra axiom (namely the Grothendieck axiom) to the theory which states the existence of those universes (it is not contradictory with the other ZFC axioms but it cant' be derived from them --- In some way the universes are too large for the ordinary ZFC axioms). The objects of Set are no longer $V$ but some universe U which solves the problem described above.

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

Note: we should add the same problem arises with other categories (such as the category whose objects are the class of all groups or the categories whose objects are the class of all topologies and so on). This is resolved the same way than category Set.

References

• 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)