The Azimuth Project
Bicategorical colimits of tensor categories

Main Results

Theorem. The 2-categories cat \mathsf{cat}_\otimes and cat fc\mathsf{cat}_{fc\otimes} of essentially small (finitely cocomplete) tensor categories together with (finitely cocontinuous) tensor functors are bicategorically cocomplete. The same holds for the corresponding variants of KK-linear structured categories cat /K\mathsf{cat}_\otimes/K and cat fc/K\mathsf{cat}_{fc\otimes}/K, where KK is any commutative ring.

Theorem. The 2-category LFP \mathsf{LFP}_\otimes, whose objects are locally finitely presentable tensor categories and whose morphisms are cocontinuous tensor functors preserving finitely presentable objects, is bicategorically cocomplete. Moreover, the inclusion LFP Cat c\mathsf{LFP}_\otimes \hookrightarrow \mathsf{Cat}_{c\otimes} into the 2-category of all cocomplete tensor categories preserves bicategorical colimits.