Pointed set

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

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

f:(X,x_{0})\to (Y,y_{0})

.

Pointed sets may be regarded as a rather simple algebraic structure. In the sense of universal algebra, they are structures with a single nullary operation which picks out the basepoint.^[7]

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

The category of pointed sets and based maps is equivalent to but not isomorphic with 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 co-slice category $\mathbf {1} \downarrow \mathbf {Set}$ , where $\mathbf {1}$ is a singleton set.^[8]^: 46^[11]

The category of pointed sets and pointed maps has both products and co-products, but it is not a distributive category. It is also an example of a category where $0\times A$ is not isomorphic to $0$ .^[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.

^ ^a ^b ^c Mac Lane (1998) p.26
^ ^a ^b Grégory Berhuy (2010). An Introduction to Galois Cohomology and Its Applications. London Mathematical Society Lecture Note Series. Vol. 377. Cambridge University Press. p. 34. ISBN 0-521-73866-0. Zbl 1207.12003.
^ ^a ^b Korte, Bernhard; Lovász, László; Schrader, Rainer (1991), Greedoids, Algorithms and Combinatorics, vol. 4, New York, Berlin: Springer-Verlag, chapter 3, ISBN 3-540-18190-3, Zbl 0733.05023
^ ^a ^b ^c Joseph Rotman (2008). An Introduction to Homological Algebra (2nd ed.). Springer Science & Business Media. ISBN 978-0-387-68324-9.
^ Maunder, C. R. F. (1996), Algebraic Topology, Dover, p. 31.
^ ^a ^b Lutz Schröder (2001). "Categories: a free tour". In Jürgen Koslowski and Austin Melton (ed.). Categorical Perspectives. Springer Science & Business Media. p. 10. ISBN 978-0-8176-4186-3.
^ Saunders Mac Lane; Garrett Birkhoff (1999) [1988]. Algebra (3rd ed.). American Mathematical Soc. p. 497. ISBN 978-0-8218-1646-2.
^ ^a ^b J. Adamek, H. Herrlich, G. Stecker, (18th January 2005) Abstract and Concrete Categories-The Joy of Cats
^ ^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.
^ 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.
^ 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.
^ ^a ^b Paolo Aluffi (2009). Algebra: Chapter 0. American Mathematical Soc. ISBN 978-0-8218-4781-7.
^ Haran, M. J. Shai (2007), "Non-additive geometry" (PDF), Compositio Mathematica, 143 (3): 618–688, MR 2330442. On p. 622, Haran writes "We consider $\mathbb {F}$ -vector spaces as ﬁnite sets $X$ with a distinguished 'zero' element..."
^ 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.

External links

[Mac26-1] Mac Lane (1998) p.26

[Berhuy-2] Grégory Berhuy (2010). An Introduction to Galois Cohomology and Its Applications. London Mathematical Society Lecture Note Series. Vol. 377. Cambridge University Press. p. 34. ISBN 0-521-73866-0. Zbl 1207.12003.

[Greedoids-3] Korte, Bernhard; Lovász, László; Schrader, Rainer (1991), Greedoids, Algorithms and Combinatorics, vol. 4, New York, Berlin: Springer-Verlag, chapter 3, ISBN 3-540-18190-3, Zbl 0733.05023

[Rotman2008-4] 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.

[KoslowskiMelton2001-6] Lutz Schröder (2001). "Categories: a free tour". In Jürgen Koslowski and Austin Melton (ed.). Categorical Perspectives. Springer Science & Business Media. p. 10. ISBN 978-0-8176-4186-3.

[LaneBirkhoff1999-7] Saunders Mac Lane; Garrett Birkhoff (1999) [1988]. Algebra (3rd ed.). American Mathematical Soc. p. 497. ISBN 978-0-8218-1646-2.

[joy-8] J. Adamek, H. Herrlich, G. Stecker, (18th January 2005) Abstract and Concrete Categories-The Joy of Cats

[LawvereSchanuel2009-9] 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.

[KoblitzZilber2009-10] 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.

[BorceuxBourn2004-11] 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.

[Aluffi2009-12] 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 $\mathbb {F}$ -vector spaces as ﬁnite sets $X$ 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.

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