Pointed set

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In mathematics, a pointed set is a set $X$ with a distinguished element $x_{0}\in X$ , which is called the basepoint. Maps of pointed sets (based maps) are those functions 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 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.

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