Pointed Set
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Pointed Set

In mathematics, a pointed set[1][2] (also based set[1] or rooted set[3]) is an ordered pair ${\displaystyle (X,x_{0})}$ where ${\displaystyle X}$ is a set and ${\displaystyle x_{0}}$ is an element of ${\displaystyle X}$ called the base point,[2] also spelled basepoint.[4]:10-11

Maps between pointed sets ${\displaystyle (X,x_{0})}$ and ${\displaystyle (Y,y_{0})}$ (called based maps,[5]pointed maps,[4] or point-preserving maps[6]) are functions from ${\displaystyle X}$ to ${\displaystyle Y}$ that map one basepoint to another, i.e. a map ${\displaystyle f\colon X\to Y}$ such that ${\displaystyle f(x_{0})=y_{0}}$. This is usually denoted

${\displaystyle f\colon (X,x_{0})\to (Y,y_{0})}$.

Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set ${\displaystyle X}$ together with a single nullary operation ${\displaystyle *:X^{0}\to X,}$ which picks out the basepoint.[7] Pointed maps are the homomorphisms of these algebraic structures.

The class of all pointed sets together with the class of all based maps form a category. In this category the pointed singleton sets ${\displaystyle (\{a\},a)}$ are initial objects and terminal objects,[1] i.e. they are zero objects.[4]:226 There is a faithful functor from pointed sets to usual sets, but it is not full and these categories are not equivalent.[8]:44 In particular, the empty set is not a pointed set because it has no element that can be chosen as the basepoint.[9]

The category of pointed sets and based maps is equivalent to the category of sets and partial functions.[6] 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."[10]

The category of pointed sets and pointed maps is isomorphic to the coslice category ${\displaystyle \mathbf {1} \downarrow \mathbf {Set} }$, where ${\displaystyle \mathbf {1} }$ is a singleton set.[8]:46[11] This coincides with the algebraic characterization, since the unique map ${\displaystyle \mathbf {1} \to \mathbf {1} }$ extends the commutative triangles defining arrows of the coslice category to form the commutative squares defining homomorphisms of the algebras.

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 ${\displaystyle 0\times A}$ is not isomorphic to ${\displaystyle 0}$.[9]

Many algebraic structures are pointed sets in a rather trivial way. For example, groups are pointed sets by choosing the identity element as the basepoint, so that group homomorphisms are point-preserving maps.[12]:24 This observation can be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets.[12]: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.[13]

As "rooted set" the notion naturally appears in the study of antimatroids[3] and transportation polytopes.[14]

## References

• Mac Lane, Saunders (1998). Categories for the Working Mathematician (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.
1. ^ a b c Mac Lane (1998) p.26
2. ^ a b Grégory Berhuy (2010). An Introduction to Galois Cohomology and Its Applications. London Mathematical Society Lecture Note Series. 377. Cambridge University Press. p. 34. ISBN 0-521-73866-0. Zbl 1207.12003.
3. ^ a b Korte, Bernhard; Lovász, László; Schrader, Rainer (1991), Greedoids, Algorithms and Combinatorics, 4, New York, Berlin: Springer-Verlag, chapter 3, ISBN 3-540-18190-3, Zbl 0733.05023
4. ^ a b c Joseph Rotman (2008). An Introduction to Homological Algebra (2nd ed.). Springer Science & Business Media. ISBN 978-0-387-68324-9.
5. ^ Maunder, C. R. F. (1996), Algebraic Topology, Dover, p. 31.
6. ^ a b Lutz Schröder (2001). "Categories: a free tour". In Jürgen Koslowski; Austin Melton (eds.). Categorical Perspectives. Springer Science & Business Media. p. 10. ISBN 978-0-8176-4186-3.
7. ^ Saunders Mac Lane; Garrett Birkhoff (1999) [1988]. Algebra (3rd ed.). American Mathematical Soc. p. 497. ISBN 978-0-8218-1646-2.
8. ^ a b J. Adamek, H. Herrlich, G. Stecker, (18 January 2005) Abstract and Concrete Categories-The Joy of Cats
9. ^ a b F. W. Lawvere; Stephen Hoel Schanuel (2009). Conceptual Mathematics: A First Introduction to Categories (2nd ed.). Cambridge University Press. pp. 296-298. ISBN 978-0-521-89485-2.
10. ^ Neal Koblitz; B. Zilber; Yu. I. Manin (2009). A Course in Mathematical Logic for Mathematicians. Springer Science & Business Media. p. 290. ISBN 978-1-4419-0615-1.
11. ^ Francis Borceux; Dominique Bourn (2004). Mal'cev, Protomodular, Homological and Semi-Abelian Categories. Springer Science & Business Media. p. 131. ISBN 978-1-4020-1961-6.
12. ^ a b Paolo Aluffi (2009). Algebra: Chapter 0. American Mathematical Soc. ISBN 978-0-8218-4781-7.
13. ^ Haran, M. J. Shai (2007), "Non-additive geometry" (PDF), Compositio Mathematica, 143 (3): 618-688, MR 2330442. On p. 622, Haran writes "We consider ${\displaystyle \mathbb {F} }$-vector spaces as finite sets ${\displaystyle X}$ with a distinguished 'zero' element..."
14. ^ Klee, V.; Witzgall, C. (1970) [1968]. "Facets and vertices of transportation polytopes". In George Bernard Dantzig (ed.). Mathematics of the Decision Sciences. Part 1. American Mathematical Soc. ASIN B0020145L2. OCLC 859802521.