Loose Yoneda theory for equipments

Abstract: While applications for double categories and proarrow equipments abound in formal category theory, categorical systems theory, and beyond, their Yoneda theory had only ever been developed for representable functors specifying tight arrows, by Paré. This is because in general, there is no ordinary representable double functor out of a (weak) double category: loose arrows would need to be sent to tight arrows (functors), while tight arrows would need to be sent to loose ones (profunctors). In recent work, Lambert, Myers, and Patterson resolve this issue by introducing twisted double functors, precisely the structure needed for a loose Hom functor. We review this construction before presenting our Yoneda lemma and embedding for these twisted representables out of double categories when they have proarrow equipments. This material is ongoing joint work with Lambert and Patterson.

category theory

Audience: researchers in the topic

Third Virtual Workshop on Double Categories