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Concept
Initial and terminal objects
Summary
In , a branch of mathematics, an initial object of a C is an object I in C such that for every object X in C, there exists precisely one morphism I → X.
The 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 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.
The empty set is the unique initial object in Set, the . 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 and every one-point space is a terminal object in this category.
In the category 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 (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 , any trivial group is a zero object. The trivial object is also a zero object in Ab, the , Rng the , R-Mod, the over a ring, and K-Vect, the over a field. See Zero object (algebra) for details. This is the origin of the term "zero object".
In Ring, the 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 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.
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