nLab terminal category

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 $1$ or $\mathbf{1}$ 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 $1$ to any category $C$ is equivalently an object of $C$. More generally, the functor category $[1, C] \simeq C$ from the terminal category to $C$ is canonically equivalent (in fact, isomorphic) to the category $C$ 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 $Cat$. 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 $1$, the functor category $[1,C]$ is equivalent, but not isomorphic, to $C$.

Proposition

Let $\mathcal{C}$ be a category.

1. The following are equivalent:

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

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

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

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

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

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

   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 $\mathcal{C}$

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

or of a terminal object

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

respectively.

Related concepts

• initial category

• point