Modular Models of Monoids with Operations
summary
A categorical framework presents equational theories of monoids equipped with arbitrary operations and shows an explicit monoid-theory correspondence: the category of algebraic-operation theories is equivalent to the category of monoids. More complex signatures (scoped or binding operations) are coreflected into algebraic operations in a way that preserves initial algebras. The paper gives concrete constructions of modular models — from monoid transformers, free algebras, composition, and symmetric monoidal categories — and shows when theories and models compose.
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chapters (13)
- 0:00Introduction
- 1:05Modeling computational effects
- 6:05Algebraic theories to monads
- 11:50Limitations of algebraic effects
- 13:39Monoids with operations in categories
- 16:55Theories with new capabilities
- 19:47Initial algebras and implementations
- 20:50Categorical families and modularity
- 21:57Semantic modularity
- 23:04Take-home message
- 23:27Q&A: Parallel composition
- 25:39Q&A: Language constructs
- 27:05Q&A: Effect handler generalization
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