nLab
cocartesian closed category

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:

Theorem

Any cocartesian closed category is equivalent to the terminal category.

Proof

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.

On the other hand, there are many co-cartesian co-closed? categories, namely the opposite category of any cartesian closed category. Any category which is both cartesian closed and co-cartesian co-closed is a preorder, though it may not be the terminal category (e.g., any Boolean algebra is such a category).

Proof

Let a category be given which is both cartesian closed and co-(cartesian closed). Cartesian closure tells us that the product of any object with the initial object 00 will be itself initial (as left adjoints preserve colimits). Furthermore, given any morphism from an object AA to 00, we can pair this with the identity function on AA to obtain a morphism from AA into A×0A \times 0 with a left inverse given by projection, thus identifying AA as a retract of an initial object, and therefore as an initial object itself. It follows immediately that any two parallel maps to 00 are equal; equivalently, by the dual reasoning, any two parallel maps out of 11 are equal. But this means any two parallel maps in general are equal, as maps from XX to YY can be identified with maps from 11 to Y XY^X; accordingly, the category must be a preorder.

Revised on September 19, 2010 10:29:13 by Sridhar Ramesh (67.180.31.171)