Beth companions of finitary essentially algebraic theories
Abstract
For categories, being balanced (meaning that every arrow that is both epic and monic is an isomorphism) can be regarded as a strong tameness property which plays an important role, for example, in algebra and logic. We study the problem of associating a balanced companion category, called a \emph{Beth companion}, to a locally finitely presentable category (equivalently, the category of models of a finitary essentially algebraic theory). We show that, if it exists, the Beth companion is unique and can be described in terms of \emph{saturated} objects. Under additional assumptions, we prove that Beth companions can be computed as orthogonality classes, and admit a syntactic presentation via Gabriel--Ulmer duality. Furthermore, we establish conditions for the transfer of properties from a category to its Beth companion. A number of examples of categories that do or do not admit a Beth companion are provided. In particular, we show that the category of commutative rings does not admit a Beth companion, and employ Pierce sheaf representations to describe the Beth companion of other classes of rings.
Related
- Notes on algebraic perspective on neural network conformal theories
- Algebraic overtwistedness of contact (+1)-surgeries via augmented Floer theories
- Discrete C*-Frobenius Algebras and Infinite-Index Extensions in Algebraic Quantum Field Theories
- A categorical model structure for generalized algebraic theories
- Derived supergeometry and perfect obstruction theories on algebraic superstacks
BibTeX
@misc{liberti2026beth,
title = {{Beth companions of finitary essentially algebraic theories}},
author = {Ivan Di Liberti and Luca Reggio},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.03601},
eprint = {2609.03601},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03601},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beth companions of finitary essentially algebraic theories
AU - Ivan Di Liberti
AU - Luca Reggio
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.03601
UR - https://arxiv.org/abs/2609.03601
AB - For categories, being balanced (meaning that every arrow that is both epic and monic is an isomorphism) can be regarded as a strong tameness property which plays an important role, for example, in algebra and logic. We study the problem of associating a balanced companion category, called a \emph{Beth companion}, to a locally finitely presentable category (equivalently, the category of models of a finitary essentially algebraic theory). We show that, if it exists, the Beth companion is unique and can be described in terms of \emph{saturated} objects. Under additional assumptions, we prove that Beth companions can be computed as orthogonality classes, and admit a syntactic presentation via Gabriel--Ulmer duality. Furthermore, we establish conditions for the transfer of properties from a category to its Beth companion. A number of examples of categories that do or do not admit a Beth companion are provided. In particular, we show that the category of commutative rings does not admit a Beth companion, and employ Pierce sheaf representations to describe the Beth companion of other classes of rings.
ER -