terminal category

The **terminal category** or **trivial category** is the terminal object in Cat. It has a single object and a single morphism. It is often denoted $1$ or $\mathbf{1}$.

A functor from $1$ to any category $C$ is the same thing as 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 $C$ 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 $C$.

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

Revised on November 4, 2009 04:55:16
by Mike Shulman
(75.3.140.104)