Discrete Lawvere theories and monads

We show that, under certain assumptions, strongly finitary enriched monads are given by discrete enriched Lawvere theories. On the other hand, monads given by discrete enriched Lawvere theories preserve surjections.

💡 Research Summary

The paper investigates the relationship between enriched Lawvere theories and monads in a symmetric monoidal closed category V under fairly general structural assumptions. Building on classic results—Linton’s correspondence between finitary Lawvere theories and finitary monads on Set, and Power’s extension to enriched settings—the author focuses on the special case where the arities are “discrete”, i.e. coproducts of the monoidal unit I. The central questions are: (1) when does a strongly μ‑ary V‑monad arise precisely from a μ‑ary discrete enriched Lawvere theory, and (2) do monads induced by discrete Lawvere theories always preserve surjections (maps f with V(I,f) surjective)?

The setting assumes that V is locally λ‑presentable and locally μ‑generated as a symmetric monoidal closed category, that the unit I is a generator, that the underlying functor V⁰(I,–) preserves λ‑presentable objects, and that (Surj, Inj) forms a V‑factorisation system. Under these hypotheses, the paper proves Theorem 3.1: strongly μ‑ary V‑monads are exactly the monads coming from μ‑ary discrete Lawvere theories. The proof proceeds by showing that a strongly μ‑ary monad T is a left Kan extension of its restriction to the dense subcategory D_μ of discrete μ‑presentable objects, hence preserves λ‑directed colimits and corresponds to a λ‑ary V‑theory. Using weighted colimit presentations of objects and the fact that T preserves such colimits, the author demonstrates that T sends any surjection to a surjection; this relies on pointwise surjectivity of the induced map ˜K(f) and a duality argument from Lack–Wood.

A crucial step is the construction of the reduct functor R: Alg(T) → Alg(T_d), where T_d is the restriction of T to the discrete subcategory D_λ. The paper shows that R is fully faithful, essentially surjective, and therefore an equivalence, using the surjectivity of the canonical maps δ_V: V⁰→V and the generator property of I. Consequently, T is completely determined by its action on discrete arities, i.e. by a μ‑ary discrete Lawvere theory.

The second main result, Theorem 4.2 (and Corollary 4.3), establishes that any monad arising from a μ‑ary discrete Lawvere theory preserves surjections. The argument uses the fact that free monads on discrete signatures are strongly μ‑ary (Proposition 4.1), and that the category of algebras for such a theory is a Birkhoff subcategory of the category of structures for the associated enriched language. Reflections in this subcategory are strong epimorphisms, which are surjective by the factorisation system, and the induced monad therefore inherits surjection preservation.

The paper illustrates the theory with examples: Pos (partial orders) satisfies the hypotheses with λ=μ=ℵ₀; Met (metric spaces with non‑expanding maps) satisfies them with λ=ℵ₁, μ=ℵ₀; and ω‑CPO also fits. In Met, the result recovers the known fact that strongly finitary monads correspond to discrete finitary Lawvere theories, while the converse direction (whether every monad from a discrete finitary theory is strongly finitary) remains open.

Overall, the work clarifies that in enriched contexts where discrete arities are available and surjections form the left class of a factorisation system, strongly μ‑ary monads and μ‑ary discrete Lawvere theories are interchangeable. Moreover, it shows that discrete Lawvere theories automatically give rise to surjection‑preserving monads, positioning them as “discrete equational theories”. These insights deepen the algebraic understanding of enriched monads and provide a robust foundation for further applications in quantitative algebra, metric semantics, and ordered algebraic structures.