On functor-quotients and their isomorphism theorems

TL;DR

This paper introduces a generalized notion of categorical quotients called $\\mathcal{F}$-quotients, which unify various quotient concepts across categories and connect algebraic and model-theoretic perspectives, leading to new isomorphism theorems.

Contribution

It develops a broader framework for quotients via functors, extending classical isomorphism theorems to new categorical and model-theoretic contexts.

Findings

Generalized isomorphism theorems for $\mathcal{F}$-quotients.

Links between categorical quotients and model-theoretic equivalence classes.

Derived quotient results for first-order structures.

Abstract

The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of $\mathcal{F}$-quotients in a category $\mathbf{C}$, which are relativized to a faithful functor $\mathcal{F}\colon \mathbf{C} \to \mathbf{D}$. The isomorphism theorems of universal algebras generalize to this setting, and we additionally find important links between $\mathcal{F}$-quotients in the concrete category of first-order structures, and quotients defined for model-theoretic equivalence classes. By first working in this categorical setting, some quotient-related results for first-order structures can be naturally obtained. In particular, we are able to prove some isomorphism theorems in the context of model theory directly from their corresponding categorical isomorphism theorems.