Initial and terminal objects

"Zero object" redirects here. For zero object in an algebraic structure, see zero object (algebra).

"Terminal element" redirects here. For the project management concept, see work breakdown structure.

In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X.

The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists a single morphism 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 I is one for which every morphism into I is an isomorphism.

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.

Morphisms of pointed sets. The image also applies to algebraic zero objects

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 $ƒ : A \to B$ 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 ( the zero 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 zero 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 fixed characteristic, the prime field is an initial object.
Any partially ordered set $(P, \leq)$ 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 $x \leq y$ . 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 $U \subset V$ . 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 zero ring) is an initial object.
If we fix a homomorphism $ƒ: A \to B$ of abelian groups, we can consider the category $C$ consisting of all pairs $(X, φ)$ where $X$ is an abelian group and $φ: X \to A$ is a group homomorphism with $ƒ φ = 0$ . A morphism from the pair $(X, φ)$ to the pair $(Y, ψ)$ is defined to be a group homomorphism $r : X \to Y$ with the property $ψ r = φ$ . 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.

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 I₁ and I₂ are two different initial objects, then there is a unique isomorphism between them. Moreover, if I is an initial object then any object isomorphic to I is also an initial object. The same is true for terminal objects.

For complete categories there is an existence theorem for initial objects. Specifically, a (locally small) complete category C has an initial object if and only if there exist a set I (not a proper class) and an I-indexed family (K_i) of objects of C such that for any object X of C there at least one morphism K_i → X for some i ∈ I.

Equivalent formulations

Terminal objects in a category C may also be defined as limits of the unique empty diagram ∅ → 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 {X_i}, in general). Dually, an initial object is a colimit of the empty diagram ∅ → C and can be thought of as an 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 objects 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 properties and adjoint functors. Let 1 be the discrete category with a single object (denoted by •), and let U : C → 1 be the unique (constant) functor to 1. Then

An initial object I in C is a universal morphism from • to U. The functor which sends • to I is left adjoint to U.
A terminal object T in C is a universal morphism from U to •. The functor which sends • to T is right adjoint to 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 X to a functor U can be defined as an initial object in the comma category (X ↓ U). Dually, a universal morphism from U to X is a terminal object in (U ↓ X).
The limit of a diagram F is a terminal object in Cone(F) the category of cones to F. Dually, a colimit of F is an initial object in the category of cones from F.
A representation of a functor F to Set is an initial object in the category of elements of F.
The notion of final functor (resp., initial functor)is a generalization of the notion of final object (resp., initial object).

Other properties

The endomorphism monoid of an initial or terminal object I is trivial: End(I) = Hom(I,I) = { id_I }.
If a category C has a zero object 0 then for any pair of objects X and Y in C the unique composition X → 0 → Y is a zero morphism from X to Y.

References

Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990). Abstract and Concrete Categories. The joy of cats (PDF). John Wiley & Sons. ISBN 0-471-60922-6. Zbl 0695.18001.
Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications 97. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.
Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.

This article is based in part on PlanetMath's article on examples of initial and terminal objects.

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