Bicategorical colimits of tensor categories

- Martin Brandenburg?, Bicategorical colimits of tensor categories, preprint, 2020. arXiv:2001.10123

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

**Theorem.** The 2-category $\mathsf{LFP}_\otimes$, 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 $\mathsf{LFP}_\otimes \hookrightarrow \mathsf{Cat}_{c\otimes}$ into the 2-category of all cocomplete tensor categories preserves bicategorical colimits.

category: mathematical methods