The Category of Factorization
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The Category of Factorization Brandon Goodell1 · Sean K. Sather-Wagstaff1 Received: 20 February 2018 / Accepted: 10 August 2020 © Springer Nature B.V. 2020
Abstract We introduce and investigate the category of factorization of a multiplicative, commutative, cancellative, pre-ordered monoid A, which we denote F (A). The objects of F (A) are factorizations of elements of A, and the morphisms in F (A) encode combinatorial similarities and differences between the factorizations. We pay particular attention to the divisibility preorder and to the monoid A = D\{0} where D is an integral domain. Among other results, we show that 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. Keywords Category · Factorization · Integral domain · Monoid · Pre-order Mathematics Subject Classification Primary: 13A05 · 06F05 · 20M50; Secondary: 13F15 · 13G05 · 20M14
Contents 1 2 3 4 5 6 7 8 9
Introduction . . . . . . . . . . . . . . . . . Definitions and Examples . . . . . . . . . . Morphisms . . . . . . . . . . . . . . . . . . Functors and Symmetric Monoidal Structure Weak Equivalences . . . . . . . . . . . . . . Weak Divisibility . . . . . . . . . . . . . . . Weak Irreducibility . . . . . . . . . . . . . . Weak Primeness . . . . . . . . . . . . . . . Factorization in Integral Domains . . . . . .
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Communicated by George Janelidze.
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Sean K. Sather-Wagstaff [email protected] https://ssather.people.clemson.edu/ Brandon Goodell [email protected]
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Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA
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B. Goodell, S. K. Sather-Wagstaff References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction It is well known that factorization in an integral domain can be poorly behaved, e.g., via the failure of unique factorization. Various constructions and techniques have been used to understand the variety of behaviors that can occur. An example of this is the group of divisibility, which goes back at least to Krull [9], and serves as important motivation for the present work. See also, e.g. [7,8,10–13]. In this paper, we introduce a construction that tracks all factorizations
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