- Category of sets
In

mathematics , the**category of sets**, denoted as**Set**, is the category whose objects are all sets and whosemorphism s are all functions. It is the most basic and the most commonly used category in mathematics.**Properties of the category of sets**The

epimorphism s in**Set**are thesurjective maps, themonomorphism s are theinjective maps, and theisomorphism s are thebijective maps.The

empty set serves as theinitial object in**Set**withempty function s as morphisms. Every singleton is aterminal object , with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus nozero object s in**Set**.The category

**Set**is complete and co-complete. The product in this category is given by thecartesian product of sets. The coproduct is given by thedisjoint 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 aconcrete 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 itspower set , and theexponential object of the sets "A" and "B" is given by the set of all functions from "A" to "B".**Set**is thus atopos (and in particular cartesian closed).**Set**is not abelian, additive or preadditive; it does not even havezero morphism s.Every

**not initial**object in**Set**is injective and (assuming theaxiom of choice ) also projective.**Foundations for the category of sets**In

Zermelo–Fraenkel set theory the collection of all sets is not a set, this follows from theaxiom of foundation . In this situation, one refers to the collection of all sets as aproper class . In ZF set theory, proper classes do not have any formal status. This is a problem: it means that the category of sets cannot be formalized straightforwardly in this setting.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 be "large".Another solution is to assume the existence of

Grothendieck universe s. 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 or its powerset will belong to the universe). The existence of universes is not implied by the usual ZF axioms; it is an additional, independent axiom. Assuming this extra axiom, the objects of**Set**can be defined to be the elements of a particular universe, rather than all the sets.Various other solutions, and variations on the above, have been proposed [

*Mac Lane, S. One universe as a foundation for category theory. Springer Lect. Notes Math. 106 (1969): 192–200.*] [*Feferman, S. Set-theoretical foundations of category theory. Springer Lect. Notes Math. 106 (1969): 201–247.*] [*Blass, A. [*] .*http://www.math.lsa.umich.edu/~ablass/interact.pdf The interaction between category theory and set theory*] . Contemporary Mathematics 30 (1984).The same issues arise with other concrete categories, such as the

category of groups or thecategory of topological spaces .**References*** (Volume 5 in the series

Graduate Texts in Mathematics )

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