The univalent foundations approach offers several insights that can be beneficial when studying monads in classical set-theoretic foundations. One of the key insights is the univalence axiom, which states that equivalent types are equal. This principle can be applied to the study of monads by allowing for a more flexible understanding of equivalences between different monadic structures. In classical set theory, one often deals with isomorphisms rather than equivalences, which can limit the ability to reason about the relationships between different monads.
By adopting a perspective informed by univalent foundations, one can leverage the notion of equivalence to establish more general results about monads, such as the ability to transfer properties between equivalent monads. This could lead to a richer understanding of how different monadic structures relate to one another, potentially revealing new connections and insights.
Additionally, the proof-relevant nature of identity types in univalent foundations can enhance the study of monads by providing a more nuanced understanding of the morphisms between monads. In classical set theory, morphisms are often treated as mere functions, but in a univalent setting, one can consider the various ways in which these morphisms can be realized, leading to a deeper exploration of the structure of monads.
Furthermore, the techniques developed in univalent foundations, such as induction on equivalences and the treatment of coherence conditions, can be applied to classical settings to simplify proofs and enhance the clarity of arguments involving monads. By incorporating these insights, researchers can develop a more unified and coherent theory of monads that bridges the gap between univalent and classical approaches, enriching the overall understanding of monadic structures in mathematics and computer science.