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
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
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