Editing Category of sets

{{Short description|Category whose objects are sets and whose morphisms are functions}}
In the [[mathematics|mathematical]] field of [[category theory]], the '''category of sets''', denoted by '''Set''', is the [[Category (mathematics)|category]] whose [[Category theory|objects]] are [[Set (mathematics)|sets]].  The arrows or [[morphism]]s between sets ''A'' and ''B'' are the [[Function (mathematics)|function]]s from ''A'' to ''B'', and the composition of morphisms is the [[Function composition|composition of functions]].

Many other categories (such as the [[category of groups]], with [[group homomorphisms]] as arrows) add structure to the objects of the category of sets or restrict the arrows to functions of a particular kind (or both).

==Properties of the category of sets==
The axioms of a category are satisfied by '''Set''' because composition of functions is [[Associative property|associative]], and because every set ''X'' has an [[identity function]] id<sub>''X''</sub> : ''X'' → ''X'' which serves as identity element for function composition.

The [[epimorphism]]s in '''Set''' are the [[surjective]] maps, the [[monomorphism]]s are the [[injective]] maps, and the [[isomorphism (category theory)|isomorphism]]s are the [[bijective]] maps.

The [[empty set]] serves as the [[initial object]] in '''Set''' with [[empty function]]s as morphisms.  Every [[singleton (mathematics)|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 object]]s in '''Set'''. 

The category '''Set''' is [[complete category|complete and co-complete]]. The [[product (category theory)|product]] in this category is given by the [[cartesian product]] of sets. The [[coproduct (category theory)|coproduct]] is given by the [[disjoint union]]: given sets ''A''<sub>''i''</sub> where ''i'' ranges over some index set ''I'', we construct the coproduct as the union of ''A''<sub>''i''</sub>×{''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 category|cartesian closed]] and [[Regular_category#Exact_%28effective%29_categories|exact in the sense of Barr]]).

'''Set''' is not [[abelian category|abelian]], [[additive category|additive]] nor [[preadditive category|preadditive]].

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

The [[Accessible category|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 [[Accessible category|locally finitely presentable category]].

If ''C'' is an arbitrary category, the [[Contravariant functor|contravariant functors]] from ''C'' to '''Set''' are often an important object of study. If ''A'' is an object of ''C'', then the functor from ''C'' to '''Set''' that sends ''X'' to Hom<sub>''C''</sub>(''X'',''A'') (the set of morphisms in ''C'' from ''X'' to ''A'') is an example of such a functor. If ''C'' is a [[Category_(mathematics)#Small_and_large_categories|small category]] (i.e. the collection of its objects forms a set), then the contravariant functors from ''C'' to '''Set''', together with natural transformations as morphisms, form a new category, a [[functor category]] known as the category of [[presheaves]] on ''C''.

==Foundations for the category of sets==

In [[Zermelo–Fraenkel set theory]] the collection of all sets is not a set; this follows from the [[axiom of foundation]]. One refers to collections that are not sets as [[proper class]]es. One cannot handle proper classes as one handles sets; in particular, one cannot write that those proper classes belong to a collection (either a set or a proper class). This is a problem because it means that the category of sets cannot be formalized straightforwardly in this setting. Categories like '''Set''' whose collection of objects forms a proper class are known as [[Large category|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 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 and its powerset will belong to the universe).  The existence of Grothendieck universes (other than the empty set and the set <math>V_\omega</math> of all [[hereditarily finite set]]s) is not implied by the usual ZF axioms; it is an additional, independent axiom, roughly equivalent to the existence of [[strongly inaccessible cardinal]]s. 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 '''Set'''<sub>''U''</sub> 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 '''Set'''<sub>''U''</sub> but not of '''Set'''.

Various other solutions, and variations on the above, have been proposed.<ref>{{harvnb|Mac Lane|1969}}</ref><ref>{{harvnb|Feferman|1969}}</ref><ref>{{harvnb|Blass|1984}}</ref>

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

== See also ==

* [[Category of topological spaces]]
* [[Set theory]]
* [[Small set (category theory)]]
* [[Category of measurable spaces]]
* [[Elementary Theory of the Category of Sets]]

==Notes==
<references/>

==References==
{{refbegin}}
*{{cite book |first=A. |last=Blass |chapter=The interaction between category theory and set theory |chapter-url=http://www.math.lsa.umich.edu/~ablass/interact.pdf |doi=10.1090/conm/030/749767 |title=Mathematical Applications of Category Theory |publisher=American Mathematical Society |series=Contemporary Mathematics |volume=30 |date=1984 |isbn=978-0-8218-5032-9 |pages=5–29 |url=}}
*{{cite book |first=S. |last=Feferman |chapter=Set-theoretical foundations of category theory |chapter-url={{GBurl|QEh8CwAAQBAJ|p=201}} |title={{harvnb|Mac Lane|1969}} |series=Lecture Notes in Mathematics |date=1969 |volume=106 |pages=201–247 |doi=10.1007/BFb0059148 |isbn=978-3-540-04625-7 }}
*Lawvere, F.W. [http://www.tac.mta.ca/tac/reprints/articles/11/tr11.pdf An elementary theory of the category of sets (long version) with commentary]
*{{cite book |first=S. |last=Mac Lane |chapter=One universe as a foundation for category theory |chapter-url= |editor-first=S. |editor-last=Mac Lane |title=Reports of the Midwest Category Seminar III |publisher=Springer |series=Lecture Notes in Mathematics |date=2006 |volume=106 |orig-date=1969 |isbn=978-3-540-36150-3 |pages=192–200 |doi=10.1007/BFb0059147 |ref={{harvid|Mac Lane|1969}}}}
* {{cite book |author-link=Saunders Mac Lane |first=Saunders |last=Mac Lane |title=Categories for the Working Mathematician |date=September 1998 |publisher=Springer |url=https://books.google.com/books?id=eBvhyc4z8HQC |isbn=0-387-98403-8 |series=[[Graduate Texts in Mathematics]] |volume=5}}
*{{Citation
| last=Pareigis
| first=Bodo
| title=Categories and functors
| year=1970
| isbn=978-0-12-545150-5
| publisher=[[Academic Press]]
| series=Pure and applied mathematics
| volume=39
}}
{{refend}}

==External links==
* [http://oeis.org/A231344 A231344&nbsp;&nbsp;&nbsp;&nbsp;Number of morphisms in full subcategories of Set spanned by {{}, {1}, {1, 2}, ..., {1, 2, ..., n}}] at [[OEIS]].

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