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.
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 $C$.
Let $\mathcal{C}$ be a category.
The following are equivalent:
$\mathcal{C}$ has a terminal object;
the unique functor $\mathcal{C} \to \ast$ to the terminal category has a right adjoint
Under this equivalence, the terminal object is identified with the image under the right adjoint of the unique object of the terminal category.
Dually, the following are equivalent:
$\mathcal{C}$ has an initial object;
the unique functor $\mathcal{C} \to \ast$ to the terminal category has a left adjoint
Under this equivalence, the initial object is identified with the image under the left adjoint of the unique object of the terminal category.
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 $\mathcal{C}$
or of a terminal object
respectively.
Last revised on April 19, 2020 at 00:16:28. See the history of this page for a list of all contributions to it.