nLab cocartesian closed category

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A cocartesian closed category is a cocartesian monoidal category which is a closed monoidal category. The notion is not very interesting, however, because of the following:


Any cocartesian closed category is equivalent to the terminal category.


Let CC be cocartesian closed. Since it has an initial object, it is inhabited; thus it suffices to show that between any two objects there is a unique morphism. But in any closed monoidal category, for any objects xx and yy the set C(x,y)C(x,y) is in bijection with C(I,[x,y])C(I,[x,y]) where II is the unit object and [,][-,-] the internal-hom. But since II is an initial object, C(I,[x,y])C(I,[x,y]) is a singleton, hence so is C(x,y)C(x,y).

Note that the proof actually applies to any semi-cocartesian closed monoidal category, i.e. any closed monoidal category whose unit is an initial object.

See also

Last revised on June 3, 2022 at 15:01:49. See the history of this page for a list of all contributions to it.