An order-theoretic perspective on categorial closure operators.

*(English)*Zbl 1406.18001The central contribution of this paper is to show how categorical closure operations (cco’s) are strictly connected with binary closure operators (bco’s), the specific instance of cco’s in posets. This fact reveals hidden links among many algebraic, topological, and categorical notions.

Intuitively, a cco is the categorical structure that abstracts over closure operators in topology, mapping a subspace to the minimal closed subspace which contains it. Conversely, a bco is an algebraic partial operation \((x,y) \mapsto x \cdot y\) on a poset, defined whenever \(x \leq y\), such that it is monotone in each argument and \(x \cdot x = x\) for every \(x\). Then \(x\) is closed in \(y\) when \(x \cdot y = x\), and dually \(x\) is dense in \(y\) when \(x \cdot y = y\).

The first result in the paper is to show that bco’s as defined before are a specialised instance of cco’s on posets. Moreover, every cco for a class \(M\) of monomorphisms in a category, under a mild condition on \(M\) together with the requirement that \(M\) is closed under composition, gives rise to a family of bco’s on posets of \(M\)-subobjects, thus allowing to deduce properties of general cco’s from the corresponding bco’s. This fact is very interesting and useful, since bco’s are simpler that cco’s, and bco’s have an intrinsic duality, which cco’s lack.

Deepening the connection, the bco’s on a poset with finite joins such that \(y\) is closed in \(z\) whenever \(x \cdot z \leq y \leq z\), are exactly those arising from unary interior operators (the dual of unary closure operators). Moreover, idempotence of a bco have several equivalent characterisations, from which one derives a strong link of bco’s with factorisation systems and torsion theory.

In the overall, the paper is apparently technical in its results, however, the clean presentation and the elegant mathematical development makes it appealing not only for the specialist, but also for mathematicians interested in the deep links among the algebra of orders, topology, and categories.

Intuitively, a cco is the categorical structure that abstracts over closure operators in topology, mapping a subspace to the minimal closed subspace which contains it. Conversely, a bco is an algebraic partial operation \((x,y) \mapsto x \cdot y\) on a poset, defined whenever \(x \leq y\), such that it is monotone in each argument and \(x \cdot x = x\) for every \(x\). Then \(x\) is closed in \(y\) when \(x \cdot y = x\), and dually \(x\) is dense in \(y\) when \(x \cdot y = y\).

The first result in the paper is to show that bco’s as defined before are a specialised instance of cco’s on posets. Moreover, every cco for a class \(M\) of monomorphisms in a category, under a mild condition on \(M\) together with the requirement that \(M\) is closed under composition, gives rise to a family of bco’s on posets of \(M\)-subobjects, thus allowing to deduce properties of general cco’s from the corresponding bco’s. This fact is very interesting and useful, since bco’s are simpler that cco’s, and bco’s have an intrinsic duality, which cco’s lack.

Deepening the connection, the bco’s on a poset with finite joins such that \(y\) is closed in \(z\) whenever \(x \cdot z \leq y \leq z\), are exactly those arising from unary interior operators (the dual of unary closure operators). Moreover, idempotence of a bco have several equivalent characterisations, from which one derives a strong link of bco’s with factorisation systems and torsion theory.

In the overall, the paper is apparently technical in its results, however, the clean presentation and the elegant mathematical development makes it appealing not only for the specialist, but also for mathematicians interested in the deep links among the algebra of orders, topology, and categories.

Reviewer: Marco Benini (Buccinasco)

##### MSC:

18A32 | Factorization systems, substructures, quotient structures, congruences, amalgams |

06A15 | Galois correspondences, closure operators (in relation to ordered sets) |

18B35 | Preorders, orders, domains and lattices (viewed as categories) |

06C05 | Modular lattices, Desarguesian lattices |

PDF
BibTeX
XML
Cite

\textit{A. M. Abdalla} and \textit{Z. Janelidze}, Quaest. Math. 41, No. 4, 529--539 (2018; Zbl 1406.18001)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Binary closure operators, Ph.D. Thesis, Stellenbosch University, South Africa, 2016 |

[2] | Cassidy, C.; Hébert, M.; Kelly, G.M., Reflective subcategories, localizations and factorization systems, Journal of Australian Mathematical Society (Series A), 38, 287-329, (1985) · Zbl 0573.18002 |

[3] | A functional approach to general topology, Categorical Foundations, Encyclopedia of Mathematics and its Applications, Vol. 97, pp. 103-163, 2004 |

[4] | Topology and its Applications 27 (1987), 129-143 |

[5] | Dikranjan, D.; Tholen, W. |

[6] | Dikranjan, D.; Tholen, W., Dual closure operators and their applications, Journal of Algebra, 439, 373-416, (2015) · Zbl 1320.18002 |

[7] | Fischer, I., Sequences of labeled trees related to Gelfand-tsetlin patterns, Advances in Applied Mathematics, 49, 165-195, (2012) · Zbl 1252.05008 |

[8] | Freyd, P.J.; Kelly, G.M., Categories of continuous functors I, Journal of Pure and Applied Algebra, 2, 169-191, (1972) · Zbl 0257.18005 |

[9] | Isbell, J.R., Algebras of uniformly continuous functions, Annals of Mathematics, 68, 2, 96-125, (1958) · Zbl 0081.11101 |

[10] | Sketches of an elephant: a topos theory compendium, Vol. 1, Oxford Logic Guides, Vol. 43, The Clarendon Press, Oxford University Press, New York, 2002 · Zbl 1071.18001 |

[11] | Korostenski, M.; Tholen, W., Factorization systems as Eilenberg-Moore algebras, Journal of Pure and Applied Algebra, 85, 57-72, (1993) · Zbl 0778.18001 |

[12] | Mac Lane, S., Duality for groups, Bulletin of the American Mathematical Society, 56, 485-516, (1950) · Zbl 0045.29905 |

[13] | Rosický, J.; Tholen, W., Factorization, fibration and torsion, Journal of Homotopy and Related Structures, 2, 2, 295-314, (2007) · Zbl 1184.18009 |

[14] | Tholen, W., Closure operators and their middle-interchange law, Topology and its Applications, 158, 2437-2441, (2011) · Zbl 1231.18003 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.