Editing Initial and terminal objects (section)

== 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.]]