Category of sets

Source: Wikipedia, the free encyclopedia.

In the

total functions from A to B, and the composition of morphisms is the composition of functions
.

Many other categories (such as the

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 axioms of a category are satisfied by Set because composition of functions is associative, and because every set X has an identity function idX : X → X which serves as identity element for function composition.

The

bijective
maps.

The

zero objects
in Set.

The category Set is

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 ensure that all the components stay disjoint).

Set is the prototype of a concrete category; other categories are concrete if they are "built on" 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 and exact in the sense of Barr).

Set is not abelian, additive nor preadditive.

Every non-empty set is an injective object in Set. Every set is a projective object in Set (assuming the axiom of choice).

The finitely presentable objects in Set are the finite sets. Since every set is a direct limit of its finite subsets, the category Set is a locally finitely presentable category.

If C is an arbitrary category, the

presheaves
on C.

Foundations for the category of sets

In

large categories
, to distinguish them from the small categories whose objects form a set.

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 of all

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 SetU 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 SetU but not of Set.

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

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

See also

Notes

  1. ^ Mac Lane 1969
  2. ^ Feferman 1969
  3. ^ Blass 1984

References

  • Blass, A. (1984). "The interaction between category theory and set theory" (PDF). Mathematical Applications of Category Theory. Contemporary Mathematics. Vol. 30. American Mathematical Society. pp. 5–29.
    ISBN 978-0-8218-5032-9
    .
  • Feferman, S. (1969). "Set-theoretical foundations of category theory".
    ISBN 978-3-540-04625-7
    .
  • Lawvere, F.W. An elementary theory of the category of sets (long version) with commentary
  • Mac Lane, S. (2006) [1969]. "One universe as a foundation for category theory". In Mac Lane, S. (ed.). Reports of the Midwest Category Seminar III. Lecture Notes in Mathematics. Vol. 106. Springer. pp. 192–200.
    ISBN 978-3-540-36150-3
    .
  • ISBN 0-387-98403-8
    .
  • Pareigis, Bodo (1970), Categories and functors, Pure and applied mathematics, vol. 39,
    ISBN 978-0-12-545150-5
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