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# Initial and terminal objects

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 IX.

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 exactly one morphism XT. 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.

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## Examples

• 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.

## 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 (Ki) of objects of C such that for any object X of C, there is at least one morphism KiX for some iI.

### Equivalent formulations

Terminal objects in a category C may also be defined as limits of the unique empty diagram 0C. 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 {Xi}, in general). Dually, an initial object is a colimit of the empty diagram 0C 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 : C1 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.

### Other properties

• The endomorphism monoid of an initial or terminal object I is trivial: End(I) = Hom(I, I) = { idI }.
• 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. Archived from the original (PDF) on 2015-04-21. Retrieved 2008-01-15.
• Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. Vol. 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. Vol. 5 (2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.