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# 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 I -> X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there exists a single morphism X -> T. 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.

### 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 schemess, 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 fieldss, 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.

• In the category of graphs, the null graph (without vertices and edges) is an initial object. The graph with a single vertex and a single 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 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 presheave.

• If we fix a homomorphism f : A -> 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 -> A 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 : X -> Y 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 : I -> C 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 : X -> F(i) is a morphism in C such that for every morphism ρ : i -> j in I, we have F(ρ)φi = φj. A morphism between pairs (X, (φi)) and (Y, (ψi)) is defined to be a morphism r : X -> Y 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.