[Paper Review] The dual of compact ordered spaces is a variety

Marco Abbadini|arXiv (Cornell University)|Feb 19, 2019

Advanced Algebra and Logic4 citations

TL;DR

This paper establishes that the dual of the category of compact ordered spaces (PosComp) is a variety of infinitary algebras, specifically denoted MC∞. Using the interval [0,1] as a dualizing object, the authors construct a dual equivalence between PosComp and MC∞ by introducing a set of finitary and countably infinite-arity operations, along with equational axioms, proving that the dual category is axiomatizable as a variety—answering an open question posed by Hofmann, Neves, and Nora (2018).

ABSTRACT

In a recent paper (2018), D. Hofmann, R. Neves and P. Nora proved that the dual of the category of compact partially ordered spaces and monotone continuous maps is a quasi-variety - not finitary, but bounded by $\aleph_1$. An open question was: is it also a variety? We show that the answer is affirmative. We describe the variety by means of a set of finitary operations, together with an operation of countably infinite arity, and equational axioms. The dual equivalence is induced by the dualizing object [0,1].

Motivation & Objective

• To resolve an open question from Hofmann, Neves, and Nora (2018) on whether the dual of compact ordered spaces is a variety.
• To provide a concrete algebraic characterization of the dual category of PosComp using operations of finite and countably infinite arity.
• To establish a dual equivalence between the category of compact ordered spaces and a variety of infinitary algebras, MC∞.
• To show that the dualizing object [0,1] induces a duality that captures all monotone continuous maps from powers of [0,1] to [0,1].
• To demonstrate that the variety MC∞ is equivalent to the category of archimedean Cauchy complete MC-algebras, thereby completing the duality.

Proposed method

• Establish a dual adjunction between the category of preordered topological spaces (PreT) and a finitary variety MC, using [0,1] as the dualizing object.
• Characterize the fixed objects of the adjunction: compact ordered spaces on the topological side and archimedean Cauchy complete MC-algebras on the algebraic side.
• Extend the finitary variety MC to an infinitary variety MC∞ by adding a countably infinite-arity operation δ, which captures limits of Cauchy sequences.
• Introduce equational axioms for MC∞ that ensure the algebraic structure reflects the topological properties of compact ordered spaces.
• Use the Subdirect Representation Theorem and an analogue of the Stone-Weierstrass Theorem to prove that the unit of the adjunction is an isomorphism for archimedean Cauchy complete MC-algebras.
• Show that MC∞ is equivalent to the category of algebras of the varietal theory whose objects are powers of [0,1] and morphisms are monotone continuous maps.

Experimental results

Research questions

• RQ1 Is the dual of the category of compact ordered spaces a variety, as opposed to just a quasi-variety?
• RQ2 Can the dual category be axiomatized using a set of operations and equational axioms, including operations of infinite arity?
• RQ3 Does the interval [0,1] serve as a dualizing object that induces a full dual equivalence between PosComp and a variety of algebras?
• RQ4 Can the class of monotone continuous maps from powers of [0,1] to [0,1] be fully captured by a finitely generated algebraic theory with infinitary operations?
• RQ5 Is the category of archimedean Cauchy complete MC-algebras equivalent to a variety of infinitary algebras?

Key findings

• The dual of the category of compact ordered spaces, PosComp^op, is equivalent to a variety of algebras, specifically the infinitary variety MC∞.
• The variety MC∞ is generated by the dualizing object [0,1], with operations including finitary operations and a countably infinite-arity operation δ that models limit-taking for Cauchy sequences.
• The dual equivalence between PosComp and MC∞ is induced by the functor Hom(−, [0,1]), and this equivalence is realized via the fixed objects of a dual adjunction.
• The category MC∞ is isomorphic to the full subcategory of archimedean Cauchy complete MC-algebras, confirming that MC∞ captures exactly the algebraic duals of compact ordered spaces.
• The operations in MC∞, when interpreted in [0,1], correspond precisely to all monotone continuous maps from any power of [0,1] to [0,1], confirming the completeness of the algebraic structure.
• The paper shows that the functor Hom(−, [0,1]) from PosComp to Set is not naturally isomorphic to the forgetful functor of a finitary variety, due to the essential dependence of the δ operation on infinitely many coordinates.