nLab
terminal object

Contents

Context

Category theory

Limits and colimits

Contents

Definition

A terminal object in a category CC is an object 11 of CC satisfying the following universal property:

for every object xx of CC, there exists a unique morphism !:x1!:x\to 1. The terminal object of any category, if it exists, is unique up to unique isomorphism. If the terminal object is also initial, it is called a zero object.

Remarks

  • Less usual synonyms are final object and terminator.

  • A terminal object is often written 11, since in Set it is a 1-element set, and also because it is the unit for the cartesian product. Other notations for a terminal object include ** and ptpt.

Properties

  • A terminal object may also be viewed as a limit over the empty diagram. Conversely, a limit over a diagram is a terminal cone over that diagram.

  • For any object xx in a category with terminal object 11, the categorical product x×1x\times 1 and the exponential object x 1x^1 both exist and are canonically isomorphic to xx.

Proposition

Let 𝒞\mathcal{C} be a category.

  1. The following are equivalent:

    1. 𝒞\mathcal{C} has a terminal object;

    2. the unique functor 𝒞*\mathcal{C} \to \ast to the terminal category has a right adjoint

      *𝒞 \ast \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \mathcal{C}

    Under this equivalence, the terminal object is identified with the image under the right adjoint of the unique object of the terminal category.

  2. Dually, the following are equivalent:

    1. 𝒞\mathcal{C} has an initial object;

    2. the unique functor 𝒞*\mathcal{C} \to \ast to the terminal category has a left adjoint

      𝒞* \mathcal{C} \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \ast

    Under this equivalence, the initial object is identified with the image under the left adjoint of the unique object of the terminal category.

Proof

Since the unique hom-set in the terminal category is the singleton, the hom-isomorphism characterizing the adjoint functors is directly the universal property of an initial object in 𝒞\mathcal{C}

Hom 𝒞(L(*),X)Hom *(*,R(X))=* Hom_{\mathcal{C}}( L(\ast) , X ) \;\simeq\; Hom_{\ast}( \ast, R(X) ) = \ast

or of a terminal object

Hom 𝒞(X,R(*))Hom *(L(X),*)=*, Hom_{\mathcal{C}}( X , R(\ast) ) \;\simeq\; Hom_{\ast}( L(X), \ast ) = \ast \,,

respectively.

Examples

Some examples of terminal objects in notable categories follow:

Last revised on June 12, 2018 at 12:35:23. See the history of this page for a list of all contributions to it.