Editing Partial function (section)

=== In category theory ===

In [[category theory]], when considering the operation of [[morphism]] composition in [[concrete categories]], the composition operation <math>\circ \;:\; \hom(C) \times \hom(C) \to \hom(C)</math> is a total function if and only if <math>\operatorname{ob}(C)</math> has one element. The reason for this is that two morphisms <math>f : X \to Y</math> and <math>g : U \to V</math> can only be composed as <math>g \circ f</math> if <math>Y = U,</math> that is, the codomain of <math>f</math> must equal the domain of <math>g.</math>

The category of sets and partial functions is [[Equivalence of categories|equivalent]] to but not [[Isomorphism of categories|isomorphic]] with the category of [[pointed set]]s and point-preserving maps.<ref name="KoslowskiMelton2001">{{cite book|editor=Jürgen Koslowski and Austin Melton|title=Categorical Perspectives|year=2001|publisher=Springer Science & Business Media|isbn=978-0-8176-4186-3|page=10|author=Lutz Schröder|chapter=Categories: a free tour}}</ref> 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>

The category of sets and partial bijections is equivalent to its [[Opposite category|dual]].<ref name="Borceux1994">{{cite book|author=Francis Borceux|title=Handbook of Categorical Algebra: Volume 2, Categories and Structures|url=https://books.google.com/books?id=5i2v9q0m5XAC&pg=PA289|year=1994|publisher=Cambridge University Press|isbn=978-0-521-44179-7|page=289}}</ref> It is the prototypical [[inverse category]].<ref name="Grandis2012">{{cite book|author=Marco Grandis|title=Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups|url=https://books.google.com/books?id=TWqhelao4KsC&pg=PA55|year=2012|publisher=World Scientific|isbn=978-981-4407-06-9|page=55}}</ref>