Core Concepts

The formal theory of monads, as established by Street, can be developed in univalent foundations. This allows for formal reasoning about various kinds of monads at the right level of abstraction.

Abstract

The paper develops the formal theory of monads in univalent foundations. It defines the bicategory of monads internal to a bicategory and proves that it is univalent. It also defines Eilenberg-Moore objects and shows that both Eilenberg-Moore categories and Kleisli categories give rise to Eilenberg-Moore objects. Finally, it relates monads and adjunctions in arbitrary bicategories.

The key highlights and insights are:

- The bicategory of monads internal to a bicategory is constructed using displayed bicategories, and it is shown to be univalent if the base bicategory is univalent.
- Monads in various bicategories, such as the bicategory of symmetric monoidal categories and the bicategory of categories with terminal objects, are characterized.
- The construction of the Kleisli category is revisited, and it is shown that in univalent foundations, every monad gives rise to an adjunction via the Kleisli category.
- The formal theory of monads provides a general setting to study various kinds of monads, including distributive laws and iterated distributive laws.
- Univalence aids the development and allows for simpler and more elegant proofs, such as induction on equivalences and the fact that certain types become mere propositions in locally univalent bicategories.

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Stats

Monads are ubiquitous in mathematics and computer science, and many different kinds of monads have been considered in various settings.
The formal theory of monads, developed by Street, provides a general setting to study various kinds of monads.
Univalent foundations allow for formal reasoning about monads at the right level of abstraction.

Quotes

"Monads are ubiquitous in both mathematics and computer science, and many different kinds of monads have been considered in various settings."
"The formal theory of monads provides a general setting to study various kinds of monads."
"Univalence also aids our development and it allows us to make several proof simpler and more elegant."

Key Insights Distilled From

by Niels van de... at **arxiv.org** 09-11-2024

Deeper Inquiries

The formal theory of monads, as developed in the context of univalent foundations, provides a robust framework for understanding various computational effects and structures in categorical semantics. In particular, monads can be utilized to model the semantics of linear logic, which emphasizes resource management and the controlled use of resources. The connection between monads and linear logic arises from the fact that linear logic can be interpreted in terms of certain types of monads that encapsulate the notion of resource consumption.
For instance, in the categorical semantics of linear logic, one can define a monad that captures the behavior of linear transformations, where the unit of the monad corresponds to the introduction of resources and the multiplication corresponds to the combination of resources. The formal theory of monads allows for the systematic study of these transformations within a bicategorical framework, enabling the exploration of various properties such as coherence and adjunctions.
Similarly, in the enriched effect calculus, which deals with effects in programming languages, the formal theory of monads can be employed to represent different computational effects such as state, exceptions, or non-determinism. By defining appropriate bicategories that encapsulate these effects, one can leverage the univalent foundations to reason about the relationships between different types of monads and their corresponding effects. This approach not only provides a unified perspective on various computational phenomena but also facilitates the formalization of these concepts in proof assistants like Coq, enhancing the rigor and reliability of the semantics.

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.

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.

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