nLab cocartesian monoidal category

Idea

A cocartesian monoidal category is a monoidal category $\big(\mathcal{C}, 0, \sqcup \big)$ whose monoidal product functor $\sqcup \,\colon\, \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is given by the coproduct (and so whose tensor unit is an initial object “$0$”).

This is the dual notion of that of a cartesian monoidal categories.

Sometime we refer to a category as cocartesian monoidal just to indicate that it has all finite coproducts.

Related concepts

• cocartesian coclosed category

• cartesian monoidal category

• cocartesian monoidal dagger category

• cocartesian monoidal preordered object

References

The terminology appears for instance in:

• Harley Eades III, Gianluigi Bellin, Def. 16 of: A Cointuitionistic Adjoint Logic [arXiv:1708.05896]