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Concept
Pointed set
Summary
In mathematics, a pointed set (also based set or rooted set) is an ordered pair where is a set and is an element of called the base point, also spelled basepoint.
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. maps such that . Based maps are 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 . In this category the pointed singleton sets 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. The base point serves as a "default value" for those arguments for which the partial function is not defined. 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 (), where is (a functor that selects) a singleton set, and (the identity functor of) the . 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.
The category of pointed sets and pointed maps has both and coproducts, but it is not a . It is also an example of a category where is not isomorphic to .
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