Pointed set

In mathematics, a pointed set (also based set or rooted set) is an ordered pair formula_1 where formula_2 is a set and formula_3 is an element of formula_2 called the base point, also spelled basepoint.
Maps between pointed sets formula_1 and formula_6 (called based maps, pointed maps, or point-preserving maps) are functions from formula_2 to formula_8 that map one basepoint to another, i.e. a map formula_9 such that formula_10. This is usually denoted
Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set formula_2 together with a single nullary operation formula_13 which picks out the basepoint. Pointed maps are the homomorphisms of these algebraic structures.
The class of all pointed sets together with the class of all based maps form a category. In this category the pointed singleton sets formula_14 are initial objects and terminal objects, i.e. they are zero objects. There is a faithful functor from pointed sets to usual sets, but it is not full and these categories are not equivalent. In particular, the empty set is not a pointed set because it has no element that can be chosen as the basepoint.
The category of pointed sets and based maps is equivalent to the category of sets and partial functions. 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."
The category of pointed sets and pointed maps is isomorphic to the coslice category formula_15, where formula_16 is a singleton set. This coincides with the algebraic characterization, since the unique map formula_17 extends the commutative triangles defining arrows of the coslice category to form the commutative squares defining homomorphisms of the algebras.
The category of pointed sets and pointed maps has both products and coproducts, but it is not a distributive category. It is also an example of a category where formula_18 is not isomorphic to formula_19.
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. This observation can be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets.
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.
As "rooted set" the notion naturally appears in the study of antimatroids and transportation polytopes.