Abstract
We introduce and investigate the category of factorization of a multiplicative, commutative, cancellative, pre-ordered monoid A, which we denote \(\mathcal {F}(A)\). The objects of \(\mathcal {F}(A)\) are factorizations of elements of A, and the morphisms in \(\mathcal {F}(A)\) encode combinatorial similarities and differences between the factorizations. We pay particular attention to the divisibility pre-order and to the monoid \(A=D{\setminus }\{0\}\) where D is an integral domain. Among other results, we show that \(\mathcal {F}(A)\) is a symmetric and strict monoidal category with weak equivalences and compute the associated category of fractions obtained by inverting the weak equivalences. Also, we use this construction to characterize various factorization properties of integral domains: atomicity, unique factorization, and so on.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, D.D., Anderson, D.F., Zafrullah, M.: Factorization in integral domains. J. Pure Appl. Algebra 69(1), 1–19 (1990)
Bénabou, Jean: Some geometric aspects of the calculus of fractions. Appl. Categ. Struct. 4(2–3), 139–165 (1996). (The European Colloquium of Category Theory (Tours, 1994))
Chapman, S., Sather-Wagstaff, S.: Irreducible divisor pair domains, preprint
Coykendall, J., Maney, J.: Irreducible divisor graphs. Commun. Algebra 35(3), 885–895 (2007)
Dwyer, W.G., Hirschhorn, P.S., Kan, D.M., Smith, J.H.: Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs, vol. 113. American Mathematical Society, Providence (2004)
Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35. Springer, New York (1967)
Gilmer, R.W.: Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., (1968)
Jaffard, P.: Les systèmes d’idéaux. Travaux et Recherches Mathématiques, IV, Dunod, Paris (1960)
Krull, W.: Allgemeine bewertungstheorie. J. Reine Angew. Math. 167, 160–196 (1932)
Lorenzen, P.: Abstrakte Begründung der multiplikativen Idealtheorie. Math. Z. 45, 533–553 (1939)
Močkoř, J.: Groups of divisibility, Mathematics and its Applications (East European Series). D. Reidel Publishing Co., Dordrecht (1983)
Ohm, J.: Semi-valuations and groups of divisibility. Canad. J. Math. 21, 576–591 (1969)
Rump, W., Yang, Y.C.: Jaffard-Ohm correspondence and Hochster duality. Bull. Lond. Math. Soc. 40(2), 263–273 (2008)
Zaks, A.: Half-factorial-domains. Israel J. Math. 37(4), 281–302 (1980)
Acknowledgements
We are grateful to Jim Coykendall and George Janelidze for their thoughtful suggestions about this work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by George Janelidze.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Goodell, B., Sather-Wagstaff, S.K. The Category of Factorization. Appl Categor Struct 28, 975–1012 (2020). https://doi.org/10.1007/s10485-020-09607-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-020-09607-9