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.
Because of Russell's paradox, which shows assuming the existence of the set of all sets leads to a contradiction, the object class of Set is a proper class, and thus the category is large.
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 initial object in Set, while every singleton is a terminal object, they are usually noted respectively 0 and 1. 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 doesn't even have zero morphisms.
Every not initial object in Set is injective and (assuming the axiom of choice) also projective.
[edit] Reference
- Mac Lane, Saunders (September 1998). Categories for the Working Mathematician. Springer. ISBN 0-387-98403-8. (Volume 5 in the series Graduate Texts in Mathematics)
