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| In the [[mathematics|mathematical]] field of [[category theory]], the '''category of sets''', denoted as '''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 all [[function (mathematics)|function]]s from ''A'' to ''B''. Care must be taken in the definition of '''Set''' to avoid [[paradoxes of set theory|set-theoretic paradoxes]].
| | Hello from Sweden. I'm glad to be here. My first name is Leilani. <br>I live in a small town called Adak in south Sweden.<br>I was also born in Adak 27 years ago. Married in September 2012. I'm working at the the office.<br><br>Also visit my blog post ... [http://tinyurl.com/n5naeyu http://tinyurl.com/n5naeyu] |
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| Many other categories (such as the [[category of groups]], with [[group homomorphisms]] as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind.
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| ==Properties of the category of sets==
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| The [[epimorphism]]s in '''Set''' are the [[surjective]] maps, the [[monomorphism]]s are the [[injective]] maps, and the [[isomorphism]]s are the [[bijective]] maps.
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| 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'''.
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| 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).
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| '''Set''' is the prototype of a [[concrete category]]; other categories are concrete if they "resemble" '''Set''' in some well-defined way.
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| 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]]).
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| '''Set''' is not [[abelian category|abelian]], [[additive category|additive]] or [[preadditive category|preadditive]]. Its [[zero morphism]]s are the empty functions ∅ → ''X''.<ref>Section I.7 of {{harvnb|Pareigis|1970}}</ref>
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| Every '''not initial''' object in '''Set''' is [[injective object|injective]] and (assuming the [[axiom of choice]]) also [[projective module|projective]].
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| ==Foundations for the category of sets==
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| 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 can't handle proper classes as one handles sets; in particular, one can't write that those proper classes belong to a collection (either a set or a proper class). This is a problem: it means that the category of sets cannot be formalized straightforwardly in this setting.
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| 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''.
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| 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''.)
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| 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'''.
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| Various other solutions, and variations on the above, have been proposed.<ref>Mac Lane 1969</ref><ref>Feferman 1969</ref><ref>Blass 1984</ref>
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| The same issues arise with other concrete categories, such as the [[category of groups]] or the [[category of topological spaces]].
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| == See also ==
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| * [[Set theory]]
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| * [[Small set (category theory)]]
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| ==Notes==
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| <references/>
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| ==References==
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| *Blass, A. [http://www.math.lsa.umich.edu/~ablass/interact.pdf The interaction between category theory and set theory]. Contemporary Mathematics 30 (1984).
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| *Feferman, S. Set-theoretical foundations of category theory. Springer Lect. Notes Math. 106 (1969): 201–247.
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| *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]
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| *Mac Lane, S. One universe as a foundation for category theory. Springer Lect. Notes Math. 106 (1969): 192–200.
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| * {{cite book |authorlink=Mac Lane|first=Saunders |last=Mac Lane |title=Categories for the Working Mathematician |date=September 1998 |publisher=Springer |url=http://books.google.com/books?id=eBvhyc4z8HQC&printsec=frontcover&cad=0#v=onepage&q&f=false |isbn=0-387-98403-8}} (Volume 5 in the series [[Graduate Texts in Mathematics]])
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| *{{Citation
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| | last=Pareigis
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| | first=Bodo
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| | title=Categories and functors
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| | year=1970
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| | isbn=978-0-12-545150-5
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| | publisher=[[Academic Press]]
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| | series=Pure and applied mathematics
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| | volume=39
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| }}
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Category_of_a_set Category of a set] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| * [http://www.encyclopediaofmath.org/index.php/Baire_theorem Baire category theorem] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| [[Category:Category-theoretic categories|Sets]]
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| [[Category:Basic concepts in set theory]]
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Hello from Sweden. I'm glad to be here. My first name is Leilani.
I live in a small town called Adak in south Sweden.
I was also born in Adak 27 years ago. Married in September 2012. I'm working at the the office.
Also visit my blog post ... http://tinyurl.com/n5naeyu