While the current development of the formal theory of monads in univalent foundations is comprehensive, it does have certain limitations. One significant limitation is the reliance on bicategories, which may not capture all the nuances of more complex categorical structures, such as higher categories or enriched categories beyond the current scope. For instance, the treatment of monads in enriched categories, where morphisms can have additional structure, may require further refinement to fully account for the interactions between the monadic structure and the enrichment.
To extend the current development, one could explore the integration of higher categorical structures, such as (∞,1)-categories, which would allow for a more flexible treatment of monads that can accommodate higher-dimensional morphisms and transformations. This could involve developing a theory of monads that is compatible with the homotopical aspects of higher categories, thereby enriching the formal theory of monads with additional layers of abstraction.
Moreover, the current framework could be expanded to include more general types of monads, such as those arising in the context of operads or monoidal categories. By establishing connections between these structures and the existing formal theory, one could create a more comprehensive framework that encompasses a wider variety of monadic phenomena, thereby enhancing the applicability of the theory across different areas of mathematics and computer science.