Editing Pointed set

{{Short description|Basic concept in set theory}}
{{Use dmy dates|date=December 2018}}
In [[mathematics]], a '''pointed set'''{{sfn|Mac&nbsp;Lane|1998}}<ref name="Berhuy">{{cite book | title=An Introduction to Galois Cohomology and Its Applications | volume=377 | series=London Mathematical Society Lecture Note Series | author=Grégory Berhuy | publisher=Cambridge University Press | year=2010 | isbn=978-0-521-73866-8 | zbl=1207.12003 | page=34 }}</ref> (also '''based set'''{{sfn|Mac&nbsp;Lane|1998}} or '''rooted set'''<ref name="Greedoids"/>) is an [[ordered pair]] <math>(X, x_0)</math> where <math>X</math> is a [[Set (mathematics)|set]] and <math>x_0</math> is an element of <math>X</math> called the '''base point''' <ref name="Berhuy"/> (also spelled '''basepoint''').<ref name="Rotman2008">{{cite book|author=Joseph Rotman|title=An Introduction to Homological Algebra|year=2008|publisher=Springer Science & Business Media|isbn=978-0-387-68324-9|edition=2nd}}</ref>{{rp|10–11}}

[[Map (mathematics)|Maps]] between pointed sets <math>(X, x_0)</math> and <math>(Y, y_0)</math>—called '''based maps''',<ref>{{citation|title=Algebraic Topology|first=C. R. F.|last=Maunder|publisher=Dover|year=1996|page=31|isbn=978-0-486-69131-2 |url=https://books.google.com/books?id=YkyizIcJdK0C&pg=PA31}}.</ref> '''pointed maps''',<ref name="Rotman2008"/> or '''point-preserving maps'''{{sfn|Schröder|2001}}—are [[function (mathematics)|functions]] from <math>X</math> to <math>Y</math> that map one basepoint to another, i.e. maps <math>f \colon X \to Y</math> such that <math>f(x_0) = y_0</math>. Based maps are usually denoted <math display=inline>f \colon (X, x_0) \to (Y, y_0)</math>.

Pointed sets are very simple [[algebraic structure]]s. In the sense of [[universal algebra]], a pointed set is a set <math>X</math> together with a single [[nullary operation]] <math>*: X^0 \to X,</math>{{efn|The notation {{math|''X''{{sup|0}}}} refers to the zeroth [[Cartesian power]] of the set {{math|''X''}}, which is a one-element set that contains the empty tuple.}} which picks out the basepoint.<ref name="LaneBirkhoff1999">{{cite book|author1=Saunders Mac Lane|author2=Garrett Birkhoff|title=Algebra|year= 1999|publisher=American Mathematical Soc.|isbn=978-0-8218-1646-2|page=497|orig-year=1988|edition=3rd}}</ref> Pointed maps are the [[homomorphism]]s of these algebraic structures.  

The [[Class (set theory)|class]] of all pointed sets together with the class of all based maps forms a [[category theory|category]].  Every pointed set can be converted to an ordinary set by forgetting the basepoint (the [[forgetful functor]] is [[faithful functor|faithful]]), but the reverse is not true.<ref name="joy">J. Adamek, H. Herrlich, G. Stecker, (18 January 2005) [http://katmat.math.uni-bremen.de/acc/acc.pdf Abstract and Concrete Categories-The Joy of Cats]</ref>{{rp|44}} In particular, the [[empty set]] cannot be pointed, because it has no element that can be chosen as the basepoint.{{sfn|Lawvere|Schanuel|2009}}

==Categorical properties==
The category of pointed sets and based maps is equivalent to the category of sets and [[partial function]]s.{{sfn|Schröder|2001}} The base point serves as a "default value" for those arguments for which the partial function is not defined. One textbook notes that "This formal completion of sets and partial maps by adding 'improper', 'infinite' elements was reinvented many times, in particular, in [[topology]] ([[one-point compactification]]) and in [[theoretical computer science]]."<ref name="KoblitzZilber2009">{{cite book|author1=Neal Koblitz|author2=B. Zilber|author3=Yu. I. Manin|title=A Course in Mathematical Logic for Mathematicians|year=2009|publisher=Springer Science & Business Media|isbn=978-1-4419-0615-1|page=290}}</ref>  This category is also isomorphic to the [[coslice category]] (<math>\mathbf{1} \downarrow \mathbf{Set}</math>), where <math>\mathbf{1}</math> is (a functor that selects) a singleton set, and <math>\scriptstyle {\mathbf{Set}}</math> (the identity functor of) the [[category of sets]].<ref name="joy"/>{{rp|46}}<ref name="BorceuxBourn2004">{{cite book|author1=Francis Borceux|author2=Dominique Bourn|title=Mal'cev, Protomodular, Homological and Semi-Abelian Categories|year=2004|publisher=Springer Science & Business Media|isbn=978-1-4020-1961-6|page=131}}</ref> This coincides with the algebraic characterization, since the unique map <math>\mathbf{1} \to \mathbf{1}</math> extends the [[Commutative diagram|commutative triangles]] defining arrows of the coslice category to form the [[Commutative diagram|commutative squares]] defining homomorphisms of the algebras.  

There is a [[faithful functor]] from pointed sets to usual sets, but it is not full and these categories are not [[Equivalence of categories|equivalent]].<ref name=joy />

The category of pointed sets is a [[pointed category]].  The pointed [[singleton set]]s <math>(\{a\}, a)</math> are both [[initial object]]s and [[terminal object]]s,{{sfn|Mac&nbsp;Lane|1998}} i.e. they are [[zero object]]s.<ref name="Rotman2008"/>{{rp|226}}  The category of pointed sets and pointed maps has both [[Product (category theory)|products]] and [[coproduct]]s, but it is not a [[distributive category]]. It is also an example of a category where <math>0 \times A</math> is not isomorphic to <math>0</math>.{{sfn|Lawvere|Schanuel|2009}}

==Applications==
Many [[variety (universal algebra)|algebraic structures]] rely on a distinguished point. For example, [[Group (mathematics)|groups]] are pointed sets by choosing the [[identity element]] as the basepoint, so that [[group homomorphism]]s are point-preserving maps.<ref name="Aluffi2009">{{cite book|author=Paolo Aluffi|title=Algebra: Chapter 0|year=2009|publisher=American Mathematical Soc.|isbn=978-0-8218-4781-7}}</ref>{{rp|24}} This observation can be restated in category theoretic terms as the existence of a [[forgetful functor]] from groups to pointed sets.<ref name="Aluffi2009"/>{{rp|582}}

A pointed set may be seen as a [[pointed space]] under the [[discrete topology]] or as a [[vector space]] over the [[field with one element]].<ref>{{citation |last=Haran |first=M. J. Shai |title=Non-additive geometry |url=http://cage.ugent.be/~kthas/Fun/library/ShaiHaran2007.pdf |journal=Compositio Mathematica |volume=143 |issue=3 |pages=618–688 |year=2007 |doi=10.1112/S0010437X06002624 |doi-broken-date=8 February 2025 |mr=2330442 |author-link=Shai Haran}}. On p.&nbsp;622, Haran writes "We consider <math>\mathbb{F}</math>-vector spaces as finite sets <math>X</math> with a distinguished 'zero' element..."</ref>

As "rooted set" the notion naturally appears in the study of [[antimatroid]]s<ref name="Greedoids">{{citation | last1=Korte | first1=Bernhard |author-link1=Bernhard Korte | last2=Lovász | first2=László | author-link2=László Lovász | last3=Schrader | first3=Rainer | year=1991 | title=Greedoids | location=New York, Berlin | publisher=[[Springer-Verlag]] | series=Algorithms and Combinatorics | volume=4 | isbn=3-540-18190-3 | zbl=0733.05023 | at=chapter 3}}</ref> and transportation polytopes.<ref>{{cite book|editor=George Bernard Dantzig|title=Mathematics of the Decision Sciences. Part 1|year=1970|orig-year=1968|publisher=American Mathematical Soc.|chapter= Facets and vertices of transportation polytopes|first1=V. | last1=Klee | first2=C. | last2=Witzgall|oclc=859802521|asin=B0020145L2}}</ref>

== See also ==
* {{annotated link|Accessible pointed graph}}
* {{annotated link|Alexandroff extension}}
* {{annotated link|Riemann sphere}}

== Notes ==
{{notelist}}

== References ==
<references />

===Further reading===
* {{cite book|first1=F.&nbsp;W.|last1=Lawvere|first2=Stephen&nbsp;Hoel|last2=Schanuel|title=Conceptual Mathematics: A First Introduction to Categories|year=2009|publisher=Cambridge University Press|isbn=978-0-521-89485-2|edition=2nd|pages=[https://archive.org/details/conceptualmathem00lawv/page/296 296–298]|url-access=registration|url=https://archive.org/details/conceptualmathem00lawv/page/296}}
* {{cite book
|last=Mac&nbsp;Lane
|first=Saunders
|author-link=Saunders Mac Lane
|title=[[Categories for the Working Mathematician]]
|publisher=Springer-Verlag
|year=1998
|edition=2nd
|isbn=0-387-98403-8
| zbl=0906.18001
}}
* {{cite book|editor-first=Jürgen|editor-last=Koslowski|editor-first2=Austin|editor-last2=Melton|title=Categorical Perspectives|year=2001|publisher=Springer Science & Business Media|isbn=978-0-8176-4186-3|page=10|author-first=Lutz|author-last=Schröder|chapter=Categories: a free tour}}

==External links==
* [https://mathoverflow.net/q/22036 Pullbacks in Category of Sets and Partial Functions]
* {{planetmath|pointedset}}
* {{nlab|id=pointed+object|title=Pointed object}}

[[Category:Basic concepts in set theory]]
[[Category:Algebraic structures]]
[[Category:Category theory]]