• The
empty set is the unique initial object in
Set, the
category of sets. Every one-element set (
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
Rel of sets and relations, the empty set is the unique initial object, the unique terminal object, and hence the unique zero object. • In the category of
pointed sets (whose objects are non-empty sets together with a distinguished element; a morphism from to being a function with ), every singleton is a zero object. Similarly, in the category of
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
integers
Z is an initial object. The
zero ring consisting only of a single element is a terminal object. • In
Rig, the category of
rigs with unity and unity-preserving morphisms, the rig of
natural numbers
N is an initial object. The zero rig, which is the
zero ring, consisting only of a single element 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 can be interpreted as a category: the objects are the elements of , and there is a single morphism from to
if and only if . This category has an initial object if and only if has a
least element; it has a terminal object if and only if has a
greatest element. •
Cat, the
category of small categories with
functors 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
schemes, Spec(
Z), the
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 of a
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 '
ChR'''
of chain complexes over a commutative ring R'', the zero complex is a zero object. • In a
short exact sequence of the form , the initial and terminal objects are the anonymous zero object. This is used frequently in
cohomology theories. • In the category of types the
empty type is the initial object and the
unit type is the terminal object. == Properties ==