## Two sides to every storey

### Ross Street – 24 November 2021

This is joint work with Branko Nikolić.

Many concepts in mathematics have relative versions. The concept applies to objects while the relative version applies to morphisms. One recaptures the original concept by specializing the domain or codomain of the relative version. Conversely, the relative version is often obtained by applying the concept to objects in a different category. In topology, proper maps relativize compactness.

More relevant to the talk is the concept of monoid in a monoidal category which is relativized by monoidal functor. A monoidal functor $T : mathcal{D}\to mathcal{C}$ can be thought of as a two-sided monoid; when $mathcal{D} = mathbf{1}$ it is a monoid in $mathcal{C}$. Yet monoidal functors are monoids in a convolution monoidal structure. This is a one-object (yet indicative) version of categories enriched in bicategories on two sides.

The plan is to revisit change of base in the sense of Kelly-Labella-Schmit-St using two-sided enrichment and to present our work with Branko on the topic. We consider there to be four results worthy of the title theorem''. We also have more examples than in our earlier preprint.

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