Maps between pointed sets and (called based maps, pointed maps, or point-preserving maps) are functions from to that map one basepoint to another, i.e. a map such that . This is usually denoted
Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set together with a single nullary operation 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 are initial objects and terminal objects, i.e. they are zero objects.:226 There is a faithful functor from pointed sets to usual sets, but it is not full and these categories are not equivalent.:44 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 , where is a singleton set.:46 This coincides with the algebraic characterization, since the unique map extends the commutative triangles defining arrows of the coslice category to form the commutative squares defining homomorphisms of the algebras.
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.:24 This observation can be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets.:582
As "rooted set" the notion naturally appears in the study of antimatroids and transportation polytopes.
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