A General Framework for Cofunctors

Bryce Clarke

Inria Saclay Centre and Macquarie University




Abstract: Cofunctors are a kind of morphism between categories which arise naturally in the context of polynomial functors. Ahman and Uustalu showed that polynomial comonads on Set correspond to categories, while the morphisms of polynomial comonads correspond to cofunctors. However, it is difficult to see how this characterisation of cofunctors might generalise to the setting of internal categories, or enriched categories, or multicategories. Moreover, it is not easily apparent how to express the duality between cofunctors and functors from the perspective of polynomial functors.

In this talk, I will develop a general framework for cofunctors as certain morphisms between monads in a double category, adapting previous work of Paré. Specialising to particular double categories will yield suitable definitions of internal cofunctor, enriched cofunctor, etc. I will also construct a pair of double categories whose objects are categories, whose horizontal morphisms are functors, and whose vertical morphisms are cofunctors, but which differ on the cells. These double categories provide a natural setting to study the interaction between cofunctors and functors, as well as to study the properties of cofunctors themselves. Finally, I will mention how this general framework for cofunctors informs the theory of (delta) lenses, and discuss some avenues for future work. The basics of double category theory will be reviewed and no prior knowledge will be required for the talk.