Editing Pointed set (section)

{{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}}