Editing Initial and terminal objects

{{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]].

== Examples ==

* The [[empty set]] is the unique initial object in '''Set''', the [[category of sets]]. Every one-element set ([[singleton (mathematics)|singleton]]) is a terminal object in this category; there are no zero objects.  Similarly, the empty space is the unique initial object in '''Top''', the [[category of topological spaces]] and every one-point space is a terminal object in this category.
* In the category '''[[Category of relations|Rel]]''' of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object.
[[Image:Terminal and initial object.svg|thumb|right|Morphisms of pointed sets. The image also applies to algebraic zero objects]]
* In the category of [[pointed set]]s (whose objects are non-empty sets together with a distinguished element; a morphism from {{math|(''A'', ''a'')}} to {{math|(''B'', ''b'')}} being a function {{math|''f'' : ''A'' → ''B''}} with {{math|1=''f''(''a'') = ''b''}}), every singleton is a zero object. Similarly, in the category of [[pointed space|pointed topological spaces]], every singleton is a zero object.
* In '''Grp''', the [[category of groups]], any [[trivial group]] is a zero object. The trivial object is also a zero object in '''Ab''', the [[category of abelian groups]], '''Rng''' the [[category of pseudo-rings]], '''''R''-Mod''', the [[category of modules]] over a ring, and '''''K''-Vect''', the [[category of vector spaces]] over a field. See ''[[Zero object (algebra)]]'' for details. This is the origin of the term "zero object".
* In '''Ring''', the [[category of rings]] with unity and unity-preserving morphisms, the ring of [[integer]]s '''Z''' is an initial object. The [[zero ring]] consisting only of a single element {{math|1=0 = 1}} is a terminal object.
* In '''Rig''', the category of [[Rig (mathematics)|rig]]s with unity and unity-preserving morphisms, the rig of [[natural number]]s '''N''' is an initial object. The zero rig, which is the [[zero ring]], consisting only of a single element {{math|1=0 = 1}} is a terminal object.
* In '''Field''', the [[category of fields]], there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, the [[prime field]] is an initial object.
* Any [[partially ordered set]] {{math|(''P'', ≤)}} can be interpreted as a category: the objects are the elements of {{math|''P''}}, and there is a single morphism from {{math|''x''}} to {{math|''y''}} [[if and only if]] {{math|''x'' ≤ ''y''}}. This category has an initial object if and only if {{math|''P''}} has a [[least element]]; it has a terminal object if and only if {{math|''P''}} has a [[greatest element]].
* '''Cat''', the [[category of small categories]] with [[functor]]s as morphisms has the empty category, '''0''' (with no objects and no morphisms), as initial object and the terminal category, '''1''' (with a single object with a single identity morphism), as terminal object.
* In the category of [[scheme (mathematics)|scheme]]s, Spec('''Z'''), the [[spectrum of a ring|prime spectrum]] of the ring of integers, is a terminal object. The empty scheme (equal to the prime spectrum of the [[zero ring]]) is an initial object.
* A [[limit (category theory)|limit]] of a [[diagram (category theory)|diagram]] ''F'' may be characterised as a terminal object in the [[category of cones]] to ''F''. Likewise, a colimit of ''F'' may be characterised as an initial object in the category of co-cones from ''F''.
* In the category '''Ch<sub>''R''</sub>''' of chain complexes over a commutative ring ''R'', the zero complex is a zero object.
* In a [[Exact sequence|short exact sequence]] of the form {{nowrap|0 → ''a'' → ''b'' → ''c'' → 0}}, the initial and terminal objects are the anonymous zero object. This is used frequently in [[Cohomology|cohomology theories.]]

== 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}}.

== References ==
* {{cite book | last1 = Adámek | first1 = Jiří | first2 = Horst | last2 = Herrlich | first3 = George E. | last3 = Strecker | year = 1990 | url = http://katmat.math.uni-bremen.de/acc/acc.pdf | title = Abstract and Concrete Categories. The joy of cats | publisher = John Wiley & Sons | isbn = 0-471-60922-6 | zbl = 0695.18001 | access-date = 2008-01-15 | archive-date = 2015-04-21 | archive-url = https://web.archive.org/web/20150421081851/http://katmat.math.uni-bremen.de/acc/acc.pdf | url-status = dead }}
* {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}
* {{cite book | first = Saunders | last = Mac Lane | author-link = Saunders Mac Lane | year = 1998 | title = [[Categories for the Working Mathematician]] | series=[[Graduate Texts in Mathematics]] | volume=5 | edition=2nd | publisher = [[Springer-Verlag]] | isbn = 0-387-98403-8 | zbl=0906.18001 }}
* ''This article is based in part on [http://www.planetmath.org PlanetMath]'s [http://planetmath.org/encyclopedia/TerminalObjectsAndZeroObjectsExamplesOfInitialObjects.html article on examples of initial and terminal objects].''

{{Category theory}}

{{DEFAULTSORT:Initial And Terminal Objects}}
[[Category:Limits (category theory)]]
[[Category:Objects (category theory)]]