Editing Initial and terminal objects (section)

== Properties ==

=== Existence and uniqueness ===

Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if {{math|''I''<sub>1</sub>}} and {{math|''I''<sub>2</sub>}} are two different initial objects, then there is a unique [[isomorphism]] between them. Moreover, if {{mvar|I}} is an initial object then any object isomorphic to {{mvar|I}} is also an initial object. The same is true for terminal objects.

For [[complete category|complete categories]] there is an existence theorem for initial objects. Specifically, a ([[locally small category|locally small]]) complete category {{mvar|C}} has an initial object if and only if there exist a set {{mvar|I}} ({{em|not}} a [[proper class]]) and an {{mvar|I}}-[[indexed family]] {{math|(''K''<sub>''i''</sub>)}} of objects of {{mvar|C}} such that for any object {{mvar|X}} of {{mvar|C}}, there is at least one morphism {{math|''K''<sub>''i''</sub> → ''X''}} for some {{math|''i'' ∈ ''I''}}.

=== Equivalent formulations ===

Terminal objects in a category {{mvar|C}} may also be defined as [[limit (category theory)|limit]]s of the unique empty [[diagram (category theory)|diagram]] {{math|'''0''' → ''C''}}. Since the empty category is vacuously a [[discrete category]], a terminal object can be thought of as an [[empty product]] (a product is indeed the limit of the discrete diagram {{math|{{mset|''X''<sub>''i''</sub>}}}}, in general). Dually, an initial object is a [[limit (category theory)|colimit]] of the empty diagram {{math|'''0''' → ''C''}} and can be thought of as an [[empty sum|empty]] [[coproduct]] or categorical sum.

It follows that any [[functor]] which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will take initial objects to initial objects. For example, the initial object in any [[concrete category]] with [[free object]]s will be the free object generated by the empty set (since the [[free functor]], being [[left adjoint]] to the [[forgetful functor]] to '''Set''', preserves colimits).

Initial and terminal objects may also be characterized in terms of [[universal property|universal properties]] and [[adjoint functors]]. Let '''1''' be the discrete category with a single object (denoted by •), and let {{math|''U'' : ''C'' → '''1'''}} be the unique (constant) functor to '''1'''. Then
* An initial object {{mvar|I}} in {{mvar|C}} is a [[universal morphism]] from • to {{mvar|U}}. The functor which sends • to {{mvar|I}} is left adjoint to ''U''.
* A terminal object {{mvar|T}} in {{mvar|C}} is a universal morphism from {{mvar|U}} to •. The functor which sends • to {{mvar|T}} is right adjoint to {{mvar|U}}.

=== Relation to other categorical constructions ===

Many natural constructions in category theory can be formulated in terms of finding an initial or terminal object in a suitable category.

* A [[universal morphism]] from an object {{mvar|X}} to a functor {{mvar|U}} can be defined as an initial object in the [[comma category]] {{math|(''X'' ↓ ''U'')}}. Dually, a universal morphism from {{mvar|U}} to {{mvar|X}} is a terminal object in {{math|(''U'' ↓ ''X'')}}.
* The limit of a diagram {{mvar|F}} is a terminal object in {{math|Cone(''F'')}}, the [[category of cones]] to {{mvar|F}}. Dually, a colimit of {{mvar|F}} is an initial object in the category of cones from {{mvar|F}}.
* A [[representable functor|representation of a functor]] {{mvar|F}} to '''Set''' is an initial object in the [[category of elements]] of {{mvar|F}}.
* The notion of [[final functor]] (respectively, initial functor) is a generalization of the notion of final object (respectively, initial object).

=== Other properties ===

* The [[endomorphism monoid]] of an initial or terminal object {{mvar|I}} is trivial: {{math|1=End(''I'') = Hom(''I'', ''I'') = {{mset| id<sub>''I''</sub> }}}}.
* If a category {{mvar|C}} has a zero object {{math|0}}, then for any pair of objects {{mvar|X}} and {{mvar|Y}} in {{mvar|C}}, the unique composition {{math|''X'' → 0 → ''Y''}} is a [[zero morphism]] from {{mvar|X}} to {{mvar|Y}}.