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 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 "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.
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—Ralph Waldo Emerson (1803–1882)
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