pointed set

In mathematics, a pointed set{{sfn|Mac Lane|1998}}{{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 }} (also based set{{sfn|Mac Lane|1998}} or rooted set) is an ordered pair $(X, x_0)$ where $X$ is a set and $x_0$ is an element of $X$ called the base point (also spelled basepoint).{{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}}{{rp|10–11}}

Maps between pointed sets $(X, x_0)$ and $(Y, y_0)$ —called based maps,{{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}}. pointed maps, or point-preserving maps{{sfn|Schröder|2001}}—are functions from $X$ to $Y$ that map one basepoint to another, i.e. maps $f \colon X \to Y$ such that $f(x_0) = y_0$ . Based maps are usually denoted $f \colon (X, x_0) \to (Y, y_0)$ .

Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set $X$ together with a single nullary operation $*: X^0 \to X,$ {{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.{{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}} Pointed maps are the homomorphisms of these algebraic structures.

The class of all pointed sets together with the class of all based maps forms a category. Every pointed set can be converted to an ordinary set by forgetting the basepoint (the forgetful functor is faithful), but the reverse is not true.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]{{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 functions.{{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."{{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}} This category is also isomorphic to the coslice category ( $\mathbf{1} \downarrow \mathbf{Set}$ ), where $\mathbf{1}$ is (a functor that selects) a singleton set, and $\scriptstyle {\mathbf{Set}}$ (the identity functor of) the category of sets.{{rp|46}}{{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}} This coincides with the algebraic characterization, since the unique map $\mathbf{1} \to \mathbf{1}$ extends the commutative triangles defining arrows of the coslice category to form the 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 equivalent.

The category of pointed sets is a pointed category. The pointed singleton sets $(\{a\}, a)$ are both initial objects and terminal objects,{{sfn|Mac Lane|1998}} i.e. they are zero objects.{{rp|226}} The category of pointed sets and pointed maps has both products and coproducts, but it is not a distributive category. It is also an example of a category where $0 \times A$ is not isomorphic to $0$ .{{sfn|Lawvere|Schanuel|2009}}

Applications

Many algebraic structures rely on a distinguished point. For example, groups are pointed sets by choosing the identity element as the basepoint, so that group homomorphisms are point-preserving maps.{{cite book|author=Paolo Aluffi|title=Algebra: Chapter 0|year=2009|publisher=American Mathematical Soc.|isbn=978-0-8218-4781-7}}{{rp|24}} This observation can be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets.{{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.{{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. 622, Haran writes "We consider $\mathbb{F}$ -vector spaces as finite sets $X$ with a distinguished 'zero' element..."

As "rooted set" the notion naturally appears in the study of antimatroids{{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}} and transportation polytopes.{{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}}

Notes

References

=Further reading=

{{cite book|first1=F. W.|last1=Lawvere|first2=Stephen 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 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