Editing Pointed set (section)

==Categorical properties==
The category of pointed sets and based maps is equivalent to the category of sets and [[partial function]]s.{{sfn|Schröder|2001}} 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]]."<ref name="KoblitzZilber2009">{{cite book|author1=Neal Koblitz|author2=B. Zilber|author3=Yu. I. Manin|title=A Course in Mathematical Logic for Mathematicians|year=2009|publisher=Springer Science & Business Media|isbn=978-1-4419-0615-1|page=290}}</ref>  This category is also isomorphic to the [[coslice category]] (<math>\mathbf{1} \downarrow \mathbf{Set}</math>), where <math>\mathbf{1}</math> is (a functor that selects) a singleton set, and <math>\scriptstyle {\mathbf{Set}}</math> (the identity functor of) the [[category of sets]].<ref name="joy"/>{{rp|46}}<ref name="BorceuxBourn2004">{{cite book|author1=Francis Borceux|author2=Dominique Bourn|title=Mal'cev, Protomodular, Homological and Semi-Abelian Categories|year=2004|publisher=Springer Science & Business Media|isbn=978-1-4020-1961-6|page=131}}</ref> This coincides with the algebraic characterization, since the unique map <math>\mathbf{1} \to \mathbf{1}</math> extends the [[Commutative diagram|commutative triangles]] defining arrows of the coslice category to form the [[Commutative diagram|commutative squares]] defining homomorphisms of the algebras.  

There is a [[faithful functor]] from pointed sets to usual sets, but it is not full and these categories are not [[Equivalence of categories|equivalent]].<ref name=joy />

The category of pointed sets is a [[pointed category]].  The pointed [[singleton set]]s <math>(\{a\}, a)</math> are both [[initial object]]s and [[terminal object]]s,{{sfn|Mac&nbsp;Lane|1998}} i.e. they are [[zero object]]s.<ref name="Rotman2008"/>{{rp|226}}  The category of pointed sets and pointed maps has both [[Product (category theory)|products]] and [[coproduct]]s, but it is not a [[distributive category]]. It is also an example of a category where <math>0 \times A</math> is not isomorphic to <math>0</math>.{{sfn|Lawvere|Schanuel|2009}}