Riguet congruences, Generalized congruences and Free monoids

We examine Riguet congruences and generalized congruences on a category, giving particular attention to their interrelations from both lattice-theoretic and category-theoretic perspectives. This investigation constitutes the principal contribution of the paper. As an application of these results, starting from a category associated with the free monoid on a set $A$ of sorts, we obtain a skeletal category via the quotient of that category by a suitable Riguet congruence on it. Moreover, we prove that this quotient category is equivalent to the category of finite $A$-sorted sets, while being neither a subcategory of it nor identical to the category of finite $A$-sorted cardinal numbers.

💡 Research Summary

The paper investigates two notions of congruence on categories—Riguet congruences, originally introduced by Riguet in the 1950s, and the more recent generalized congruences introduced by Bednarczyk et al.—and establishes a detailed relationship between them from both lattice‑theoretic and categorical perspectives.

A Riguet congruence Φ on a category C consists of an equivalence relation Φ_ob on objects together with a family of relations Φ_fl on morphisms between equivalent objects. The definition includes six compatibility conditions: (a) identities of equivalent objects are related, (b) the diagonal of each hom‑set is contained in the corresponding Φ_fl, (c) symmetry under swapping source and target, (d) vertical composition compatibility, (e) horizontal composition compatibility, and (f) a “choice” condition guaranteeing that for any morphism f: a→b and any equivalent objects a′, b′ there exists a morphism f′: a′→b′ with (f,f′)∈Φ_fl. These conditions ensure that the quotient category C/Φ is well defined and that the canonical projection pr_Φ: C → C/Φ is a functor.

The authors prove that the collection RCgr(C) of all Riguet congruences on a fixed category C, ordered by inclusion, forms a bounded directed‑complete poset. In particular there is a smallest congruence (the diagonal) and a largest one (the total relation). This shows that Riguet congruences enjoy a natural lattice‑like structure, although not a complete lattice in general.

Generalized congruences extend the idea by allowing equivalence not only on individual morphisms but also on non‑empty finite sequences of composable morphisms. The set GCgr(C) of all generalized congruences, ordered by inclusion, is shown to be an algebraic lattice: compact elements correspond to congruences generated by finitely many basic identifications, and every congruence is the directed join of such compact ones.

A key contribution is the introduction of “strong generalized congruences”. For each Riguet congruence Φ the authors construct a canonical strong generalized congruence Φ♮. This construction yields a Scott‑continuous map (·)♮ : RCgr(C) → GCgr(C) which preserves directed joins and sends the minimal Riguet congruence to the minimal strong generalized congruence. The authors give an explicit description of the morphism component Φ♮_fl as the smallest strong generalized congruence containing Φ_fl.

From the categorical side, the authors define two classification categories: RCCat, whose objects are pairs (C,Φ) with Φ a Riguet congruence, and GCCat, whose objects are pairs (C,Ψ) with Ψ a strong generalized congruence. The map (·)♮ lifts to a functor (·)♮ : RCCat → GCCat. Moreover, they construct natural transformations pr_R : UR → QR, pr_G : UG → QG (where UR, UG are the forgetful functors to Cat and QR, QG are the quotient functors) and a natural transformation α : QR → QG∘(·)♮. Under the existence of a compatible choice function for the hom‑sets, α becomes an equivalence, showing that the quotient by a Riguet congruence and the quotient by its associated strong generalized congruence are equivalent categories.

The second part of the paper applies this theory to many‑sorted free monoids. For a set A of sorts, let A⋆ denote the free monoid of finite words over A. The authors introduce an equivalence relation ≡_A on A⋆ that identifies two words whenever they have the same multiset of letters (i.e., the same A‑sorted cardinalities). They then construct a canonical permutation for each pair of ≡_A‑equivalent words, which serves to define the morphism component of a Riguet congruence Φ on the category C(A⋆) associated with the free monoid.

The quotient category Q(A⋆) = C(A⋆)/Φ is proved to be skeletal. Its objects correspond precisely to A‑sorted finite cardinal numbers (families (n_a)_{a∈A} of natural numbers with only finitely many non‑zero entries). The authors define a functor |·|_a : Q(A⋆) → Set_A^f that sends each object to the corresponding A‑sorted finite set and each morphism to the induced function between such sets. This functor is shown to be an equivalence of categories, establishing that Q(A⋆) is equivalent to the category Set_A^f of finite A‑sorted sets.

Importantly, Q(A⋆) is not a subcategory of Set_A^f, because its objects are not actual sets but rather cardinality data; nor is it identical to the skeletal subcategory Card_A^f of finite A‑sorted cardinal numbers, since the morphisms in Q(A⋆) arise from the canonical permutations rather than arbitrary functions.

Finally, the authors prove functoriality of the constructions: the assignment A ↦ C(A⋆) and A ↦ Q(A⋆) are functors Set → Cat, and the family of projections pr_{≡A} forms a natural transformation C ⇒ Q. Moreover, for any function f : A ⇉ B, there is a natural isomorphism MSet′(f)∘|·|_A ≅ |·|_B∘Q(f), showing that the equivalence respects change of sorts.

In summary, the paper provides a thorough comparative study of Riguet and generalized congruences, establishes a lattice‑theoretic bridge via a Scott‑continuous map, and demonstrates the utility of this bridge by constructing a new skeletal category equivalent to finite many‑sorted sets but distinct from the usual cardinal‑number skeleton. This contributes both to the theory of congruences on categories and to the categorical modeling of many‑sorted algebraic structures.