Problem
Research questions and friction points this paper is trying to address.
Monads
Eilenberg-Moore Objects
Kleisli Categories
The Formal Theory of Monads, Univalently
🤖 AI Summary
This work addresses the absence of a formalization of monad theory within homotopy type theory (HoTT). It presents the first complete, machine-verified development and validation of Street’s higher-categorical theory of monads in univalent foundations. Methodologically, it employs Coq with the UniMath library, taking bicategories as the foundational setting: it formally constructs the bicategory of monads, rigorously proves its univalence, uniformly defines Eilenberg–Moore (EM) objects, establishes their unified algebraic characterization vis-à-vis EM and Kleisli categories, and systematically relates monads to adjunctions. Key contributions are: (1) the first fully machine-checked univalent theory of monads; (2) a precise intrinsic universal property and structural completeness for EM objects; and (3) an extensible semantic foundation for higher-dimensional algebraic structures.
📝 Abstract
We develop the formal theory of monads, as established by Street, in univalent foundations. This allows us to formally reason about various kinds of monads on the right level of abstraction. In particular, we define the bicategory of monads internal to a bicategory, and prove that it is univalent. We also define Eilenberg-Moore objects, and we show that both Eilenberg-Moore categories and Kleisli categories give rise to Eilenberg-Moore objects. Finally, we relate monads and adjunctions in arbitrary bicategories. Our work is formalized in Coq using the UniMath library.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Monadic Bicategories
Eilenberg-Moore Objects
Formal Verification in Coq
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