Initial and Terminal Objects - Examples

Examples

• The empty set is the unique initial object in 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 the category of topological spaces; 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 zero object.
• In the category of non-empty sets, there are no initial objects. The singletons are not initial: while every non-empty set admits a function from a singleton, this function is in general not unique.

• 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 ƒ : A → B with ƒ(a) = b), every singleton is a zero object. Similarly, in the category of pointed topological spaces, every singleton is a zero object.
• In the category of semigroups, the empty semigroup is the unique initial object and any singleton semigroup is a terminal object. There are no zero objects. In the subcategory of monoids, however, every trivial monoid (consisting of only the identity element) is a zero object.
• In the category of groups, any trivial group is a zero object. There are zero objects also for the category of abelian groups, category of pseudo-rings Rng (trivial ring), category of modules over a ring, and category of vector spaces over a field; see zero object (algebra) for details. This is the origin of the term "zero object".
• In the category of rings with unity and unity-preserving morphisms, the ring of integers Z is an initial object. The trivial ring consisting only of a single element 0=1 is a terminal object.
• In the category of fields, there are no initial or terminal objects. However, in the subcategory of fields of characteristic p the prime field of characteristic p forms 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 y if and only if x ≤ 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.
• All monoids may be considered, in their own right, to be categories with a single object. In this sense, each monoid is a category that consists of one object and a collection of specific morphisms to itself. This one object is neither initial or terminal unless the monoid is trivial, in which case it is both.
• In the category of graphs, the null graph, containing no vertices nor edges, is an initial object. If loops are permitted, then the graph with a single vertex and one loop is terminal. The category of simple graphs does not have a terminal object.
• Similarly, the category of all small categories with functors as morphisms has the empty category as initial object and the category 1 (with a single object and morphism) as terminal object.
• Any topological space X can be viewed as a category by taking the open sets as objects, and a single morphism between two open sets U and V if and only if U ⊂ V. The empty set is the initial object of this category, and X is the terminal object. This is a special case of the case "partially ordered set", mentioned above. Take P :=the set of open subsets.
• If X is a topological space (viewed as a category as above) and C is some small category, we can form the category of all contravariant functors from X to C, using natural transformations as morphisms. This category is called the category of presheaves on X with values in C. If C has an initial object c, then the constant functor which sends every open set to c is an initial object in the category of presheaves. Similarly, if C has a terminal object, then the corresponding constant functor serves as a terminal presheaf.
• 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 trivial ring) is an initial object.
• If we fix a homomorphism ƒ : A → B of abelian groups, we can consider the category C consisting of all pairs (X, φ) where X is an abelian group and φ : X → A is a group homomorphism with . A morphism from the pair (X, φ) to the pair (Y, ψ) is defined to be a group homomorphism r : X → Y with the property . The kernel of ƒ is a terminal object in this category; this is nothing but a reformulation of the universal property of kernels. With an analogous construction, the cokernel of ƒ can be seen as an initial object of a suitable category.
• In the category of interpretations of an algebraic model, the initial object is the initial algebra, the interpretation that provides as many distinct objects as the model allows and no more.