Category Of Sets

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 Epimorphism s in Set are the Surjective maps, the Monomorphism s are the Injective maps, and the Isomorphism s are the Bijective maps.

The Empty Set serves as Initial Object in Set, while every Singleton is a Terminal Object . There are thus no Zero Object s 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 doesn't even have Zero Morphism s.

Every not initial object in '''Set''' is Injective and (assuming the Axiom Of Choice ) also Projective .

	CATEGORIES ABOUT CATEGORY OF SETS

	category-theoretic categories

	sets

	basic concepts in set theory

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