Pointed set

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In mathematics, a pointed set[1][2] (also based set[1] or rooted set[3]) is an ordered pair ${\displaystyle (X,x_{0})}$ where ${\displaystyle X}$ is a set and ${\displaystyle x_{0}}$ is an element of ${\displaystyle X}$ called the base point,[2] also spelled basepoint.[4]:10–11

Maps between pointed sets ${\displaystyle (X,x_{0})}$ and ${\displaystyle (Y,y_{0})}$ (called based maps,[5] pointed maps,[4] or point-preserving maps[6]) are functions from ${\displaystyle X}$ to ${\displaystyle Y}$ that map one basepoint to another, i.e. a map ${\displaystyle f:X\to Y}$ such that ${\displaystyle f(x_{0})=y_{0}}$. This is usually denoted

${\displaystyle f:(X,x_{0})\to (Y,y_{0})}$.

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.[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 ${\displaystyle (\{a\},a)}$ 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 ${\displaystyle \mathbf {1} \downarrow \mathbf {Set} }$, where ${\displaystyle \mathbf {1} }$ 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]

References

1. Mac Lane (1998) p.26
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8. J. Adamek, H. Herrlich, G. Stecker, (18th January 2005) Abstract and Concrete Categories-The Joy of Cats
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13. {{#invoke:citation/CS1|citation |CitationClass=citation }}. On p. 622, Haran writes "We consider ${\displaystyle \mathbb {F} }$-vector spaces as ﬁnite sets ${\displaystyle X}$ with a distinguished ‘zero’ element..."
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