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 or or .
In enriched category theory, often instead of the terminal category one is interested in the unit enriched category.
A functor from the terminal category to any category is equivalently an object of . More generally, the functor category from the terminal category to is canonically equivalent (in fact, isomorphic) to the category 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 (). 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 . 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 , the functor category is equivalent, but not isomorphic, to .
Let be a category.
The following are equivalent:
has a terminal object;
the unique functor 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:
has an initial object;
the unique functor 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
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.