In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in, there exists precisely one morphism. The dual notion is that of a terminal object : is terminal if for every object in there exists exactly one morphism. 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 is one for which every morphism into is an isomorphism.
Examples
The empty set is the unique initial object in Set, the category of sets. Every one-element set 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, every singleton is a zero object. Similarly, in the category of pointed topological spaces, every singleton is a zero 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 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.
Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if and are two different initial objects, then there is a unique isomorphism between them. Moreover, if is an initial object then any object isomorphic to 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 complete category has an initial object if and only if there exist a set and an -indexed family of objects of such that for any object of, there is at least one morphism for some.
Equivalent formulations
Terminal objects in a category may also be defined as limits of the unique empty diagram. Since the empty category is vacuously a discrete category, a terminal object can be thought of as an empty product. Dually, an initial object is a colimit of the empty diagram 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. 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, and let be the unique functor to 1. Then
A terminal object in is a universal morphism from to •. The functor which sends • to is right adjoint to.
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 to a functor can be defined as an initial object in the comma category. Dually, a universal morphism from to is a terminal object in.
The limit of a diagram is a terminal object in, the category of cones to. Dually, a colimit of is an initial object in the category of cones from.
