Operadic-like categorical description of PROPs and properads

Ascertain whether fully-fledged PROPs or at least properads can be realized as operad-type objects governed by an operadic-like category, and, if so, construct the appropriate categorical framework and demonstrate the correspondence.

Background

The Epilogue proposes extending the bivariant operadic framework to structures generalizing PROPs, suggesting ‘contramodules’ as a potential direction. The authors pose a broader question about whether PROPs or properads themselves admit an operad-type description via an operadic-like category.

They explicitly state that they do not know how to achieve this and invite suggestions, leaving a substantial open problem in the categorical foundations of PROPs/properads.