Strongly finitary metric monads are too strong

Varieties of quantitative algebras are fully described by their free-algebra monads on the category Met of metric spaces. For a longer time it has been an open problem whether the resulting enriched monads are precisely the strongly finitary ones (determined by their values on finite discrete spaces). We present a counter-example: the variety of algebras on two close binary operations yields a monad which is not strongly finitary. A full characterization of free-algebra monads of varieties is: they are the 1-basic monads, i.e., weighted colimits of strongly finitary monads (in the category of enriched finitary monads). As a consequence, strongly finitary endofunctors on Met are not closed under composition.

💡 Research Summary

The paper addresses a long‑standing open problem in the theory of quantitative algebras: whether every free‑algebra monad arising from a variety of quantitative algebras on the category Met of (extended) metric spaces is necessarily strongly finitary. Strongly finitary endofunctors, in the sense of Kelly and Lack, are those obtained by left Kan‑extending their restriction to the full subcategory of finite discrete spaces. While all known examples (e.g., the n‑th power functor, the Hausdorff monad, product with a fixed space) are strongly finitary, it has been conjectured that this property holds for all free‑algebra monads.

The authors first recall the enriched categorical setting: Met is a symmetric monoidal closed category with the sum metric on the cartesian product, and CMet (its full subcategory of complete spaces) is reflective via Cauchy completion. An endofunctor is finitary if it preserves directed colimits; it is strongly finitary if it is the left Kan extension of its values on finite discrete spaces. Theorem 2.11 gives a concrete characterisation: a functor T is strongly finitary iff (a) it is finitary, (b) for every space X the map T i_X (where i_X:|X|→X is the canonical inclusion of the underlying set) is an epimorphism, and (c) any map f: T|X|→Y satisfying the ε‑condition (2.1) factors uniquely through T i_X.

The paper then reviews quantitative algebras: a Σ‑algebra equipped with a metric such that each operation is non‑expansive with respect to the maximum metric on product spaces. Free quantitative algebras T_Σ X are built from term algebras equipped with a recursively defined distance d*_X. The associated monad T_Σ is strongly finitary (Corollary 3.5) and preserves epimorphisms.

The central contribution is a counter‑example. The authors define a variety V consisting of algebras with two binary operations ⊕ and ⊗ that are ε‑close: for all elements a,b we require d(⊕(a,b), ⊗(a,b)) ≤ ε for a fixed ε>0. This is a 1‑basic variety, i.e., it is presented by a single quantitative equation of the form t = ε t′. They show that the free‑algebra monad T_V fails condition (b) of Theorem 2.11: the map T_V i_X is not an epimorphism for certain X, and the ε‑condition does not guarantee a factorisation. Consequently T_V is not strongly finitary. This demonstrates that the class of strongly finitary monads is strictly smaller than the class of free‑algebra monads of varieties.

Motivated by this, the authors introduce the notion of a “1‑basic monad”. A monad is 1‑basic if it can be expressed as a weighted colimit (in the 2‑category of enriched finitary monads Mnd_f(Met)) of strongly finitary monads. Weighted colimits are built from diagrams indexed by a weight W: D→Met; in Met every space X can be written as a weighted colimit of a diagram of discrete spaces (its “foliation”), where the indexing category is the real interval (0,∞) together with two extra cones. Theorem 1.1 proves that a monad arises as the free‑algebra monad of some variety of quantitative algebras iff it is 1‑basic. Thus the free‑algebra monads are precisely the weighted colimits of strongly finitary monads.

A striking corollary (Theorem 1.2) is that strongly finitary endofunctors on Met are not closed under composition. The proof uses the counter‑example: the composite of two strongly finitary functors can yield the non‑strongly‑finitary monad T_V. This contradicts a conjecture suggested by Bourke and Garner’s theory of saturated classes of arities, which would have implied closure under composition if finite discrete spaces formed a saturated class in Met.

The paper also extends the analysis to the complete setting CMet. The same characterisation holds: free‑algebra monads on CMet are exactly the weighted colimits of strongly finitary monads on CMet. Many familiar monads (e.g., the Hausdorff monad) remain strongly finitary, while the ε‑close binary‑operation variety still provides a non‑strongly‑finitary example. Moreover, varieties consisting only of unary operations always yield strongly finitary monads (Section 6).

In the related work discussion, the authors compare their results with earlier attempts (e.g., the incomplete arguments in