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Combinatorics of factorization systems on lattices

Jishnu Bose, Tien Chih, Hannah Housden, Legrand Jones, Chloe Lewis, Kyle Ormsby, Millie Rose

TL;DR

We develop a combinatorial framework for factorization systems on finite lattices by constructing characteristic and cocharacteristic endomorphisms $\chi^{(L,R)}$ and $\lambda^{(L,R)}$ that encode how $0\to x$ and $x\to 1$ factor. We prove these maps are antitone with images corresponding to interior and closure operators, and we establish that fibers over $\lambda$ are intervals whose endpoints are reflective/coreflective factorization systems, respectively; this yields bijections between (co)reflective factorization systems and certain operator classes. The work reveals deep connections to transfer systems, submonoids (Moore families), and closure structures, and situates factorization systems as a unifying language across combinatorics and equivariant homotopy theory, with consequences for $N_\infty$ operads and related model structures. Furthermore, a network of dualities and Galois connections shows how (co)reflective systems encode ubiquitous combinatorial data, including poly-Bernoulli enumerations in modular lattices, and suggests broad applicability to homotopy-theoretic frameworks.

Abstract

We initiate the combinatorial study of factorization systems on finite lattices, paying special attention to the role that reflective and coreflective factorization systems play in partitioning the poset of factorization systems on a fixed lattice. We ultimately uncover an intricate web of relations with such diverse combinatorial structures as submonoids, monads, Moore systems, transfer systems (from stable equivariant homotopy theory), and poly-Bernoulli numbers.

Combinatorics of factorization systems on lattices

TL;DR

We develop a combinatorial framework for factorization systems on finite lattices by constructing characteristic and cocharacteristic endomorphisms and that encode how and factor. We prove these maps are antitone with images corresponding to interior and closure operators, and we establish that fibers over are intervals whose endpoints are reflective/coreflective factorization systems, respectively; this yields bijections between (co)reflective factorization systems and certain operator classes. The work reveals deep connections to transfer systems, submonoids (Moore families), and closure structures, and situates factorization systems as a unifying language across combinatorics and equivariant homotopy theory, with consequences for operads and related model structures. Furthermore, a network of dualities and Galois connections shows how (co)reflective systems encode ubiquitous combinatorial data, including poly-Bernoulli enumerations in modular lattices, and suggests broad applicability to homotopy-theoretic frameworks.

Abstract

We initiate the combinatorial study of factorization systems on finite lattices, paying special attention to the role that reflective and coreflective factorization systems play in partitioning the poset of factorization systems on a fixed lattice. We ultimately uncover an intricate web of relations with such diverse combinatorial structures as submonoids, monads, Moore systems, transfer systems (from stable equivariant homotopy theory), and poly-Bernoulli numbers.

Paper Structure

This paper contains 6 sections, 17 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Factorization and transfer systems on lattices
  3. Factorization systems
  4. Transfer systems
  5. Characteristic and cocharacteristic functions
  6. The unbearable ubiquity of (co)reflective factorization systems

Key Result

Theorem 1

Let $P$ be a finite lattice. Then there are inverse poset isomorphisms

Figures (2)

  • Figure 1: Relations invoked in the proof of \ref{['lemma:fiberclosejoin']}.
  • Figure 2: Above the dashed line: Bijections between reflective factorization systems and other mathematical structures. Below the dashed line: The same, but for coreflective factorization systems. Each arrow is labeled by the number of the theorem (or proposition, etc.) establishing the bijection. The dashed line represents the formal duality between factorization systems $(L, R)$ on $P$ and $(R^{\mathrm{op}}, L^{\mathrm{op}})$ on $P^{\mathrm{op}}$.

Theorems & Definitions (45)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1: fooqw
  • Definition 5
  • Definition 6
  • Definition 7: REUChar
  • Theorem 2: REUChar
  • Definition 8
  • ...and 35 more