nLab
terminal category

Contents

Contents

Definition

The terminal category or trivial category or final category is the terminal object in Cat. It is the unique (up to isomorphism) category with a single object and a single morphism, necessarily the identity morphism on that object. It is often denoted 11 or 1\mathbf{1} or *\ast.

In enriched category theory, often instead of the terminal category one is interested in the unit enriched category.

Properties

A functor from the terminal category 11 to any category CC is the equivalently an object of CC. More generally, the functor category [1,C]C[1, C] \simeq C from the terminal category to CC is canonically equivalent (in fact, isomorphic) to the category CC itself.

The terminal category is a discrete category that, as a set, may be called the singleton. As a subset of the singleton, it is in fact a truth value, true (\top). In general, all of these (and their analogues in higher category theory and homotopy theory) may be called the point.

So far we have interpreted “terminal” as referring to the 1-category CatCat. If instead we interpret “terminal” in the 2-categorical sense, then any category equivalent to the one-object-one-morphism category described above is also terminal. A category is terminal in this sense precisely when it is inhabited and indiscrete. For such a category 11, the functor category [1,C][1,C] is equivalent, but not isomorphic, to CC.

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.

Last revised on June 13, 2018 at 11:36:18. See the history of this page for a list of all contributions to it.