Free monoids and Riguet congruences
Juan Climent Vidal, Enric Cosme Llópez, Raúl Ruiz Mora
TL;DR
The paper develops skeletal models for categories arising from free monoids by constructing the category $oldsymbol{C}(oldsymbol{A}^{oldsymbol{*}})$ and its skeletal quotient $oldsymbol{Q}(oldsymbol{A}^{oldsymbol{*}})$ via a Riguet congruence. It proves that $oldsymbol{C}(oldsymbol{A}^{oldsymbol{*}})$ is equivalent to the finite multi-sorted Set category $ extsf{Set}^{A}_{ ext{f}}$, and that $oldsymbol{Q}(oldsymbol{A}^{oldsymbol{*}})$ is skeletal and equivalent to $ extsf{Set}^{A}_{ ext{f}}$ as well as to $oldsymbol{C}(oldsymbol{A}^{oldsymbol{*}})/oldsymbol{≡^{A^{ atural}}}$. The work then situates Riguet congruences inside a broader lattice-theoretic and category-theoretic framework, introducing strong generalized congruences and an isotone, Scott-continuous map from $ ext{RCgr}(oldsymbol{C})$ to $ ext{GCgr}(oldsymbol{C})$, with adjunctions between corresponding classified categories. In the key case $oldsymbol{C}=oldsymbol{C}(oldsymbol{A}^{oldsymbol{*}})$, the authors produce a skeletal quotient $oldsymbol{Q}(oldsymbol{A}^{oldsymbol{*}})$ that is equivalent to $ extsf{Set}^{A}_{ ext{f}}$ and to the quotient by the strong generalized congruence, thereby unifying algebraic and categorical perspectives on quotients of free monoid-based structures.
Abstract
We begin by associating to $\mathbf{A}^{\star}$, the free monoid on a set $A$, a category $\mathsf{C}(\mathbf{A}^{\star})$ -- an instance of the free coproduct completion of a discrete category -- which is in general non-skeletal, and by proving that it is equivalent to $\mathsf{Set}^{A}_{\mathrm{f}}$, the category of finite $A$-sorted sets. Next, as a step toward constructing a skeletal quotient category of $\mathsf{C}(\mathbf{A}^{\star})$ via the notion of a Riguet congruence on a category, we recall this notion, correct and complete it, and examine its relationship with generalized congruences from both lattice-theoretic and category-theoretic perspectives. In particular, after introducing the notion of strong generalized congruence on a category, we prove that, for any category $\mathsf{C}$, there exists an isotone and Scott continuous morphism from $(\mathrm{RCgr}(\mathsf{C}),\subseteq)$, the bounded directed-complete ordered set of Riguet congruences on $\mathsf{C}$ to $(\mathrm{GCgr}(\mathsf{C}),\subseteq)$, the algebraic lattice of generalized congruences on $\mathsf{C}$, that sends a Riguet congruence $Φ$ on $\mathsf{C}$ to the strong generalized congruence $Φ^{\natural}$ on $\mathsf{C}$. Finally, for a suitable Riguet congruence on $\mathsf{C}(\mathbf{A}^{\star})$, denoted by $\equiv^{A}$, we construct a skeletal quotient category $\mathsf{Q}(\mathbf{A}^{\star})$ of $\mathsf{C}(\mathbf{A}^{\star})$ and prove that it is equivalent to $\mathsf{Set}^{A}_{\mathrm{f}}$ and also to $\mathsf{C}(\mathbf{A}^{\star})/{\equiv^{A\natural}}$, where $\equiv^{A\natural}$ is the strong generalized congruence on $\mathsf{C}(\mathbf{A}^{\star})$ canonically associated to $\equiv^{A}$.