Semi-Strongly Finitary Monads

• Semi-strongly finitary monads are defined via weighted colimits in Met_f, providing canonical presentations of free-algebra monads for quantitative algebraic varieties.
• They overcome limitations of strongly finitary monads by relaxing strict factorization properties, as demonstrated by explicit counterexamples in metric spaces.
• Their construction ensures stability under weighted colimits and enrichment-tensors, facilitating robust applications in metric-enriched universal algebra.

A semi-strongly finitary monad is a specific class of Met-enriched monads characterized by their role as canonical presentations of free-algebra monads for varieties of quantitative algebras on the category Met of (extended) metric spaces and nonexpansive maps. These monads are defined via weighted colimits of strongly finitary monads, providing a full characterization of all free-algebra monads of quantitative algebraic varieties. The concept arises from the realization that strongly finitary monads—previously conjectured to capture these varieties—are in fact too restrictive, as evidenced by explicit counterexamples. The framework of weighted colimits in the enriched category Met_f (of finitary Met-enriched monads and monad morphisms) is fundamental to this theory and its closure properties.

1. Formal Definition and Category-Theoretic Context

Let Met denote the symmetric monoidal closed category of (extended) metric spaces and nonexpansive maps. A Met-enriched monad $T$ on Met is said to be finitary if its underlying endofunctor preserves directed colimits ([Definition 2.1], (Adamek, 6 Jan 2026)). It is strongly finitary if it is constructed as the left Kan extension of its restriction to finite discrete spaces ([Definition 2.4]), equivalently, if $T$ preserves directed colimits, the components $T i_X$ are surjective for all $X$, and $T$ satisfies the neighborhood-factorization property ([Proposition 2.5]).

A semi-strongly finitary Met-enriched monad $T$ is defined by the existence of a weighted colimit expression:

$T \cong \operatorname{colim}_W D$

for an enriched diagram $D:\mathcal{J} \to \mathrm{Met}_f$ and a weight $W:\mathcal{J}^{op} \to [0,\infty]$, in the enriched category Met_f ([Definition of semi-strongly finitary], (Adamek, 6 Jan 2026)). Here, $\mathrm{Met}_f$ refers to the category of finitary Met-enriched monads and morphisms. The notion of weighted colimit employed is that of the universal solution to

$$\mathrm{Met}_f(T, -) \cong [\mathcal{J}^{op}, \mathrm{Met}_f](W, \mathrm{Met}_f(D-, -))$$

as is standard in enriched category theory.

2. Main Characterization Theorem

The crucial structural result is the Main Theorem ([Theorem 5.1], (Adamek, 6 Jan 2026)):

Let $T$ be a Met-enriched monad on Met (or on CMet, the complete spaces). $T$ is the free-algebra monad $T_V$ of some variety $V$ of quantitative algebras if and only if $T$ is semi-strongly finitary.

Varieties $V$ are defined as full subcategories of the category $\Sigma$–Met (for some finitary signature $\Sigma$) presented by a set of quantitative ($\epsilon$-) equations $t =_{\epsilon} t'$ for $\epsilon \geq 0$. The free-algebra monad $T_V$ thus exists, is enriched, and finitary ([Theorem 3.9], [Corollary 3.11]).

3. Construction via Weighted Colimits

Every variety $V$ has a presentation by generators (signature $\Sigma$) and an (possibly infinite) set of quantitative equations $E = \{ s_i =_{\epsilon_i} t_i \}_{i \in I}$. For each such equation, introduce a “one-operation signature” $\Gamma_i$ with a single $n(i)$-ary symbol $\gamma_i$. Form two strongly finitary monads:

$$T_{\Gamma_i} \xrightarrow{\ \widehat{s_i}\ } T_\Sigma,\quad T_{\Gamma_i} \xrightarrow{\ \widehat{t_i}\ } T_\Sigma$$

where $T_\Sigma$ denotes the free-$\Sigma$-algebra monad ([Example 6.2(2)]). The monad for $V$ is constructed as the weighted colimit (specifically, the coequalizer) of $T_{\Gamma_i} \rightrightarrows T_\Sigma$ weighted by the two-point pseudometric $\{\square, \lozenge\}$ with $d(\square,\lozenge) = \epsilon_i$ ([Proposition 4.2], [Remark 4.3]). Thus, variety monads $T_V$ always arise as weighted colimits of strongly finitary building blocks $T_\Sigma$ and $T_{\Gamma_i}$.

The diagram below summarizes the construction:

Component	Role	Notes
$T_\Sigma$	Strongly finitary free algebra monad (generators)	Index object (weight 0)
$T_{\Gamma_i}$	Strongly finitary for equation $i$	Weight: two-point pseudometric ($\epsilon_i$)
Weighted colimit (coeq.)	Form $T_V$ as colimit of above	Encodes the full set of equations $E$

4. Limitations of Strongly Finitary Monads and Counterexamples

The hypothesis that all free-algebra monads of varieties are strongly finitary fails. The explicit counterexample ([Section 8]) uses the signature $\Sigma = \{\sigma_1, \sigma_2\}$ (two binary operations) and imposes the single quantitative equation $\sigma_1(x,y) =_c \sigma_2(x,y)$ with $0 < c < \infty$. In this case, the corresponding free-algebra monad $T_V$ acts as:

$X \mapsto (T_\Sigma|X|, \hat{d}_X)$

where $\hat{d}_X$ is the maximal metric making leaves nonexpansive and penalizing root-label changes by $+c$. This $T_V$ is not strongly finitary, as it fails the neighborhood-factorization property ([Proposition 2.5]), even though every variety monad is always semi-strongly finitary.

5. Closure Properties

Semi-strongly finitary monads exhibit several closure properties ([Theorem 4.5]):

• Closed under weighted colimits in $\mathrm{Met}_f$ (including ordinary coproducts and coequalizers).
• Closed under enrichment-tensors (tensoring with any metric space $M$).
• Stable under retracts in $\mathrm{Met}_f$.
• All free-algebra monads $T_V$ preserve surjections and filtered colimits ([Corollary 3.11]).

6. Illustrative Examples

• Unary varieties: If $\Sigma$ has only nullary or unary symbols, every $V$-equation is of depth at most one. In this case, $T_V$ satisfies the factorization condition and is strongly finitary ([Section 6]).
• Classical (ordinary) equations ($\epsilon = 0$): For varieties with only classical equations, one constructs a pseudometric $d^{@}_X$ on $T_\Sigma|X|$ as the chain-infimum of free-algebra distances differing at most by equations. Again, $T_V$ is strongly finitary ([Section 7]).
• General varieties: The opposite category $\mathrm{Var(Met)}^{op}$ embeds fully and faithfully in $\mathrm{Met}_f$ via $V \mapsto T_V$ ([Proposition 4.4]). Since $\mathrm{Var(Met)}^{op}$ is enriched-cocomplete, its dual is enriched-complete, ensuring every variety monad arises as a weighted colimit of generating strongly finitary monads.

7. Significance and Related Structures

The framework of semi-strongly finitary monads enables the systematic algebraic treatment of varieties of quantitative algebras on metric spaces, overcoming the incompleteness of the strongly finitary hypothesis. It provides an explicit means to build all free-algebra monads for such varieties via weighted colimits of elementary, strongly finitary pieces. This result refines the categorical landscape of metric-enriched universal algebra and establishes new limits for the expressiveness of strongly finitary constructions in this context (Adamek, 6 Jan 2026).