# Initial object

In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism IX. The dual notion is that of a terminal object: T is terminal, if to every object X in C there exists a single morphism XT. Initial objects are also called coterminal and terminal objects are also called final. If an object is both initial and terminal, we call it a zero object.

## Properties

Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an initial object exists, then it is unique up to a unique isomorphism. The same is true for terminal objects.

## 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 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 f : AB 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 the category of groups, any trivial group (consisting only of its identity element) is a zero object. The same is true for the category of abelian groups as well as for the category of left modules over a fixed ring. This is the origin of the term "zero object".
• In the category of rings, the ring of integers (and any ring isomorphic to it) serves as an initial object. The trivial ring consisting only of a single element 0=1 is a terminal object.
• In the category of schemes, the prime spectrum of Z is a terminal object. The empty scheme (equal to the prime spectrum of the trivial ring) is an initial object.
• In the category of fields, there are no initial or terminal objects.
• 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 xy. This category has an initial object if and only if P has a smallest element; it has a terminal object if and only if P has a largest element. This explains the terminology.
• Similarly, the category of all small categories with functors as morphisms has the empty category as initial object and the one-object-one-morphism category 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 UV. The empty set is the initial object of this category, and X is the terminal object.
• 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.
• If we fix a homomorphism f : AB of abelian groups, we can consider the category C consisting of all pairs (X, φ) where X is an abelian group and φ : XA is a group homomorphism with f φ = 0. A morphism from the pair (X, φ) to the pair (Y, ψ) is defined to be a group homomorphism r : XY with the property ψ r = φ. The kernel of f 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 f can be seen as an initial object of a suitable category.
• We can treat arbitrary limits of functors similar to the previous example: if F : IC is a functor, we define a new category Cone(F) as follows: its objects are pairs (X, (φi)) where X is an object of C and for every object i of I, φi : XF(i) is a morphism in C such that for every morphism ρ : ij in I, we have F(ρ)φi = φj. A morphism between pairs (X, (φi)) and (Y, (ψi)) is defined to be a morphism r : XY such that ψi r = φi for all objects i of I. The universal property of the limit can then be expressed as saying: any terminal object of Cone(F) is a limit of F and vice versa (note that Cone(F) need not contain a terminal object, just like F need not have a limit).
• Generalizing the previous two examples: every construction described by a universal property can be reformulated as the problem of finding an initial or terminal object in a suitable category.

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