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 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. i.e., An may be regarded as a where is the one-point set. Pointed maps are the homomorphisms of these algebras.
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, i.e. a zero object.: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, for it has no element that can be chosen as base point.
The category of pointed sets and based maps is equivalent to but not isomorphic with 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 co-slice 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
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