nLab
monoidally cocomplete category

A monoidally cocomplete category is a (small-) cocomplete category CC bearing a monoidal category structure, such that the monoidal product \otimes is cocontinuous in each of its separate variables, i.e., XX \otimes - is cocontinuous for each XX and Y- \otimes Y is cocontinuous for each YY. 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 CC, the cocontinuity in separate variables of the monoidal product implies that CC is monoidal biclosed. For instance, if CC is locally presentable or is total, then being symmetric monoidally cocomplete is equivalent to being symmetric monoidal closed.

Created on August 8, 2013 at 21:39:12. See the history of this page for a list of all contributions to it.