# Pointed set

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

Maps between pointed sets and (called **based maps**,^{[5]} **pointed maps**,^{[4]} or **point-preserving maps**^{[6]}) are functions from to that map one basepoint to another, i.e. a map such that . This is usually denoted

- .

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 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 , where 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 is not isomorphic to .^{[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]}

## See also

## References

- Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*(2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.

- 1 2 3 Mac Lane (1998) p.26
- 1 2 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. - 1 2 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 - 1 2 3 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. - 1 2 Lutz Schröder (2001). "Categories: a free tour". In Jürgen Koslowski and Austin Melton.
*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. - 1 2 J. Adamek, H. Herrlich, G. Stecker, (18th January 2005) Abstract and Concrete Categories-The Joy of Cats
- 1 2 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. - 1 2 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 -vector spaces as ﬁnite sets with a distinguished 'zero' element..." - ↑ Klee, V.; Witzgall, C. (1970) [1968]. "Facets and vertices of transportation polytopes". In George Bernard Dantzig.
*Mathematics of the Decision Sciences. Part 1*. American Mathematical Soc. ASIN B0020145L2. OCLC 859802521.

## External links

- http://mathoverflow.net/questions/22036/pullbacks-in-category-of-sets-and-partial-functions
- Pointed set at PlanetMath.org.
- Pointed object in
*nLab*