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In mathematics, a pointed setcite-ref-footnotemac-nbsp-lane1998-1-0[1]cite-ref-berhuy-2-0[2] (also based setcite-ref-footnotemac-nbsp-lane1998-1-1[1] or rooted setcite-ref-greedoids-3-0[3]) is an ordered pair ( X , x 0 ) {\displaystyle (X,x_{0})} where X {\displaystyle X} is a set and x 0 {\displaystyle x_{0}} is an element of X {\displaystyle X} called the base point cite-ref-berhuy-2-1[2] (also spelled basepoint).cite-ref-rotman2008-4-0[4]

Maps between pointed sets ( X , x 0 ) {\displaystyle (X,x_{0})} and ( Y , y 0 ) {\displaystyle (Y,y_{0})} —called based maps,cite-ref-5[5] pointed maps,cite-ref-rotman2008-4-1[4] or point-preserving mapscite-ref-footnoteschr-der2001-6-0[6]—are functions from X {\displaystyle X} to Y {\displaystyle Y} that map one basepoint to another, i.e. maps f : : X → → Y {\displaystyle f\colon X\to Y} such that f ( x 0 ) = y 0 {\displaystyle f(x_{0})=y_{0}} . Based maps are usually denoted f : : ( X , x 0 ) → → ( Y , y 0 ) {\textstyle 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 {\displaystyle X} together with a single nullary operation ∗ ∗ : X 0 → → X , {\displaystyle *:X^{0}\to X,} cite-ref-7[a] which picks out the basepoint.cite-ref-lanebirkhoff1999-8-0[7] 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.cite-ref-joy-9-0[8] In particular, the empty set cannot be pointed, because it has no element that can be chosen as the basepoint.cite-ref-footnotelawvereschanuel2009-10-0[9]

• Categorical properties

• Applications

• See also

• Notes

• References

• Further reading

• External links

The category of pointed sets and based maps is equivalent to the category of sets and partial functions.cite-ref-footnoteschr-der2001-6-1[6] 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-ref-koblitzzilber2009-11-0[10] This category is also isomorphic to the coslice category ( 1 ↓ ↓ S e t {\displaystyle \mathbf {1} \downarrow \mathbf {Set} } ), where 1 {\displaystyle \mathbf {1} } is (a functor that selects) a singleton set, and S e t {\displaystyle \scriptstyle {\mathbf {Set} }} (the identity functor of) the category of sets.cite-ref-joy-9-1[8]cite-ref-borceuxbourn2004-12-0[11] This coincides with the algebraic characterization, since the unique map 1 → → 1 {\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.

There is a faithful functor from pointed sets to usual sets, but it is not full and these categories are not equivalent.cite-ref-joy-9-2[8]

The category of pointed sets is a pointed category. The pointed singleton sets ( { a } , a ) {\displaystyle (\{a\},a)} are both initial objects and terminal objects,cite-ref-footnotemac-nbsp-lane1998-1-2[1] i.e. they are zero objects.cite-ref-rotman2008-4-2[4] 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 × × A {\displaystyle 0\times A} is not isomorphic to 0 {\displaystyle 0} .cite-ref-footnotelawvereschanuel2009-10-1[9]

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-ref-aluffi2009-13-0[12] This observation can be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets.cite-ref-aluffi2009-13-1[12]

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.cite-ref-14[13]

As "rooted set" the notion naturally appears in the study of antimatroidscite-ref-greedoids-3-1[3] and transportation polytopes.cite-ref-15[14]

• Accessible pointed graph

• Alexandroff extension – Way to extend a non-compact topological space

• Riemann sphere – Model of the extended complex plane plus a point at infinity

cite-note-7a. ↑ The notation X0 refers to the zeroth Cartesian power of the set X, which is a one-element set that contains the empty tuple.

cite-note-footnotemac-nbsp-lane1998-11. ↑ Mac Lane 1998.

cite-note-berhuy-22. ↑ citerefgr-gory-berhuy2010Grégory Berhuy (2010). An Introduction to Galois Cohomology and Its Applications. London Mathematical Society Lecture Note Series. Vol. 377. Cambridge University Press. p. 34. ISBN 978-0-521-73866-8. Zbl 1207.12003.

cite-note-greedoids-33. ↑ citerefkortelov-szschrader1991Korte, 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

cite-note-rotman2008-44. ↑ citerefjoseph-rotman2008Joseph Rotman (2008). An Introduction to Homological Algebra (2nd ed.). Springer Science & Business Media. ISBN 978-0-387-68324-9.

cite-note-55. ↑ citerefmaunder1996Maunder, C. R. F. (1996), Algebraic Topology, Dover, p. 31, ISBN 978-0-486-69131-2.

cite-note-footnoteschr-der2001-66. ↑ Schröder 2001.

cite-note-lanebirkhoff1999-87. ↑ citerefsaunders-mac-lanegarrett-birkhoff1999Saunders Mac Lane; Garrett Birkhoff (1999) [1988]. Algebra (3rd ed.). American Mathematical Soc. p. 497. ISBN 978-0-8218-1646-2.

cite-note-joy-98. ↑ J. Adamek, H. Herrlich, G. Stecker, (18 January 2005) Abstract and Concrete Categories-The Joy of Cats

cite-note-footnotelawvereschanuel2009-109. ↑ Lawvere & Schanuel 2009.

cite-note-koblitzzilber2009-1110. ↑ citerefneal-koblitzb-zilberyu-i-manin2009Neal 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.

cite-note-borceuxbourn2004-1211. ↑ citereffrancis-borceuxdominique-bourn2004Francis Borceux; Dominique Bourn (2004). Mal'cev, Protomodular, Homological and Semi-Abelian Categories. Springer Science & Business Media. p. 131. ISBN 978-1-4020-1961-6.

cite-note-aluffi2009-1312. ↑ citerefpaolo-aluffi2009Paolo Aluffi (2009). Algebra: Chapter 0. American Mathematical Soc. ISBN 978-0-8218-4781-7.

cite-note-1413. ↑ citerefharan2007Haran, M. J. Shai (2007), "Non-additive geometry" (PDF), Compositio Mathematica, 143 (3): 618–688, doi:10.1112/S0010437X06002624, MR 2330442. On p. 622, Haran writes "We consider F {\displaystyle \mathbb {F} } -vector spaces as finite sets X {\displaystyle X} with a distinguished 'zero' element..."

cite-note-1514. ↑ citerefkleewitzgall1970Klee, 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.

• citereflawvereschanuel2009Lawvere, F. W.; Schanuel, Stephen Hoel (2009). Conceptual Mathematics: A First Introduction to Categories (2nd ed.). Cambridge University Press. pp. 296–298. ISBN 978-0-521-89485-2.

• citerefmac-lane1998Mac Lane, Saunders (1998). Categories for the Working Mathematician (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.

• citerefschr-der2001Schröder, Lutz (2001). "Categories: a free tour". In Koslowski, Jürgen; Melton, Austin (eds.). Categorical Perspectives. Springer Science & Business Media. p. 10. ISBN 978-0-8176-4186-3.

• Pullbacks in Category of Sets and Partial Functions

• Pointed set at PlanetMath.

• Pointed object at the nLab