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Calculating monad transformers with category theory

Oleksandr Manzyuk

TL;DR

This work shows that state, writer, reader, and error monad transformers can be uniformly obtained by translating a monad along an adjunction, encapsulated by the construction P = UTF. By instantiating the general translation to specific adjunctions (e.g., F = -×S, U = (-)^S for state; F = X×M, U forgetful for writer; environment-based reader; and E/Set for error), it derives concrete transformer monads and their unit and multiplication from the adjunction and the base monad. The approach unifies several transformers under a single categorical mechanism and connects to strength and distributive-law ideas, while noting that continuation monad transformers fall outside this framework. The results provide both a principled theoretical lens and practical translation rules that illuminate how these transformers arise from adjunctions and lifted monads.

Abstract

We show that state, reader, writer, and error monad transformers are instances of one general categorical construction: translation of a monad along an adjunction.

Calculating monad transformers with category theory

TL;DR

This work shows that state, writer, reader, and error monad transformers can be uniformly obtained by translating a monad along an adjunction, encapsulated by the construction P = UTF. By instantiating the general translation to specific adjunctions (e.g., F = -×S, U = (-)^S for state; F = X×M, U forgetful for writer; environment-based reader; and E/Set for error), it derives concrete transformer monads and their unit and multiplication from the adjunction and the base monad. The approach unifies several transformers under a single categorical mechanism and connects to strength and distributive-law ideas, while noting that continuation monad transformers fall outside this framework. The results provide both a principled theoretical lens and practical translation rules that illuminate how these transformers arise from adjunctions and lifted monads.

Abstract

We show that state, reader, writer, and error monad transformers are instances of one general categorical construction: translation of a monad along an adjunction.

Paper Structure

This paper contains 6 sections, 1 theorem, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Translating monads along adjunctions
  3. State monad transformer
  4. Writer monad transformer
  5. Reader monad transformer
  6. Error monad transformer

Key Result

Proposition 2.2

Let $(F, U, \eta, \varepsilon)$ be an adjunction from $\mathop{\mathrm{\mathcal{C}}}\nolimits$ to $\mathop{\mathrm{\mathcal{D}}}\nolimits$. Suppose that $(T, e^T, m^T)$ is a monad on the category $\mathop{\mathrm{\mathcal{D}}}\nolimits$. Then the functor $P = UTF : \mathop{\mathrm{\mathcal{C}}}\noli is a monad on the category $\mathop{\mathrm{\mathcal{C}}}\nolimits$.

Figures (2)

  • Figure 1: Proof of the identity axioms for the monad $P$.
  • Figure 2: Proof of the associativity axiom for the monad $P$.

Theorems & Definitions (9)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 5.1
  • Remark 5.2