Initial and terminal objects

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In category theory, an abstract 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 a single 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.

Contents

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 groups, any trivial group is a zero object. The same is true for the categories of abelian groups, modules over a ring, and vector spaces over a field. This is the origin of the term "zero object".
  • In the category of semigroups, the empty semigroup the unique initial object and any singleton semigroup is a terminal object. There are no zero objects. In the subcategory of monoids, however, every trivial monoid (consisting of only the identity element) is a 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 : 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 rings with unity and unity-preserving morphisms, the ring of integers Z is an initial object. The trivial ring consisting only of a single element 0=1 is a terminal object. In the category of general rings with homomorphisms, the trivial ring is a zero object.
  • In the category of fields, there are no initial or terminal objects. However, in the subcategory of fields of characteristic p, the prime field of characteristic p forms an initial object.
  • 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 least element; it has a terminal object if and only if P has a greatest element.
  • If a monoid is considered as a category with a single object, this object is neither initial or terminal unless the monoid is trivial, in which case it is both.
  • 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 category 1 (with a single object and morphism) 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. This is a special case of the case "partially ordered set", mentioned above. Take P:= the set of open subsets
  • 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.
  • In the category of schemes, Spec(Z) the prime spectrum of the ring of integers is a terminal object. The empty scheme (equal to the prime spectrum of the trivial ring) is an initial object.
  • 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.
  • In the category of interpretations of an algebraic model, the initial object is the initial algebra, the interpretation that provides as many distinct objects as the model allows and no more.

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