Pointed set
In mathematics, a pointed set is an ordered pair where is a set and is an element of called the basepoint.[1]
Maps of between pointed sets and (based maps) 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.
The class of all pointed sets together with the class of all based maps form a category.
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.
There is a faithful functor from usual sets to pointed sets, but it is not full, and these categories are not equivalent.
References
- 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.
- Mac Lane, Saunders (1998). Categories for the Working Mathematician (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8.
- ^ Pointed Set at PlanetMath