Pointed set

In mathematics, a pointed set is an ordered pair $(X,x_{0})$ where $X$ is a set and $x_{0}$ is an element of $X$ called the basepoint.

Maps of between pointed sets $(X,x_{0})$ and $(Y,y_{0})$ (based maps) 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.

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.