nLab probicategory

Idea

A probicategory is a “many-object” generalisation of a promonoidal category, in the same way that a bicategory is a “many-object” generalisation of a monoidal category.

Just as a promonoidal category in the most general context in which we can perform Day convolution to obtain a monoidal structure on the category of presheaves, a probicategory is the most general context in which we can perform convolution locally to obtain a bicategory structure on the local cocompletion.

Definition

A probicategory is a bicategory enriched in the monoidal bicategory Prof.

Properties

Given a probicategory, we may form a bicategory by change of base along the pseudofunctor from Prof to Cat given by taking free cocompletion. When applied to a one-object probicategory (i.e. a promonoidal category), this produces the (monoidal) presheaf category obtained through Day convolution.

Related pages

• promonoidal category
• bicategory
• Day convolution

References

• Brian Day, Construction of Biclosed Categories (PhD Thesis)

A summary of the results of the thesis may be found in:

• Brian Day, Biclosed bicategories: localisation of convolution, arXiv:0705.3485