Editing Initial and terminal objects (section)

{{short description|Special objects used in (mathematical) category theory}}
{{redirect|Zero object|zero object in an algebraic structure|zero object (algebra)}}
{{redirect|Terminal element|the project management concept|work breakdown structure}}
In [[category theory]], a branch of [[mathematics]], an '''initial object''' of a [[category (mathematics)|category]] {{mvar|C}} is an object {{mvar|I}} in {{mvar|C}} such that for every object {{mvar|X}} in {{mvar|C}}, there exists precisely one [[morphism]] {{math|''I'' → ''X''}}.

The [[dual (category theory)|dual]] notion is that of a '''terminal object''' (also called '''terminal element'''): {{mvar|T}} is terminal if for every object {{mvar|X}} in {{mvar|C}} there exists exactly one morphism {{math|''X'' → ''T''}}. Initial objects are also called '''coterminal''' or '''universal''', and terminal objects are also called '''final'''.

If an object is both initial and terminal, it is called a '''zero object''' or '''null object'''.  A '''pointed category''' is one with a zero object.

A [[strict initial object]] {{mvar|I}} is one for which every morphism into {{mvar|I}} is an [[isomorphism]].