A *monoidally cocomplete category* is a (small-) cocomplete category $C$ bearing a monoidal category structure, such that the monoidal product $\otimes$ is cocontinuous in each of its separate variables, i.e., $X \otimes -$ is cocontinuous for each $X$ and $- \otimes Y$ is cocontinuous for each $Y$. The term is due to Max Kelly.

Similarly, one has the notions of symmetric monoidally cocomplete category, braided monoidally cocomplete category, cartesian monoidally cocomplete category, and so on.

Under nice conditions on the category $C$, the cocontinuity in separate variables of the monoidal product implies that $C$ is monoidal biclosed. For instance, if $C$ is locally presentable or is total, then being symmetric monoidally cocomplete is equivalent to being symmetric monoidal closed.

Any cocomplete cartesian closed category is monoidally cocomplete.

Day convolution provides a construction of the free monoidally cocomplete category on a small monoidal category.

Last revised on December 15, 2022 at 20:37:39. See the history of this page for a list of all contributions to it.