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 11 or 1\mathbf{1}.

A functor from 11 to any category CC is the same thing as 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 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.

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

Last revised on November 4, 2009 at 04:55:16. See the history of this page for a list of all contributions to it.