nLab monoidally cocomplete category

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 closed monoidal category is monoidally cocomplete.

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

Related pages

• monoidal category
• cocomplete category
• Day convolution

References

• Geun Bin Im and G. M. Kelly, A universal property of the convolution monoidal structure, J. Pure Appl. Algebra 43 (1986), no. 1, 75-88.