Terminal object

Examples

• In the category of pointed sets (whose objects are non-empty sets together with a distinguished element; a morphism from (A, a) to (B, b) being a function f : A → B with f(a) = b), 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 integersZ is an initial object. The zero ring consisting only of a single element 0 = 1 is a terminal object.
• In Rig, the category of rigs with unity and unity-preserving morphisms, the rig of natural numbersN is an initial object. The zero rig, which is the zero ring, consisting only of a single element 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(P, ≤) can be interpreted as a category: the objects are the elements of P, and there is a single morphism from x to yif and only ifx ≤ y. This category has an initial object if and only if P has a least element; it has a terminal object if and only if P 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 diagramF 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_R of chain complexes over a commutative ring R, the zero complex is a zero object.
• In a short exact sequence of the form 0 → a → b → c → 0, 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.