# 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.^{[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

- ↑
^{1.0}^{1.1}^{1.2}Mac Lane (1998) p.26 - ↑
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^{8.0}^{8.1}J. Adamek, H. Herrlich, G. Stecker, (18th January 2005) Abstract and Concrete Categories-The Joy of Cats - ↑
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^{12.0}^{12.1}{{#invoke:citation/CS1|citation |CitationClass=book }} - ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}. On p. 622, Haran writes "We consider -vector spaces as ﬁnite sets with a distinguished ‘zero’ element..."
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