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**Connectedness in a category.**
*(English)*
Zbl 1446.18002

There exists a convenient categorical generalization of the concept of closure operator in a topological space (see, for example, [D. Dikranjan and W. Tholen, Categorical structure of closure operators. With applications to topology, algebra and discrete mathematics. Dordrecht: Kluwer Academic Publishers (1995; Zbl 0853.18002)] for a thorough description of the topic). More precisely, a closure operator \(C\) in a category X with respect to a class \(\mathcal{M}\) of X-subobjects is given by a family \(C=(c_X)_{X\in Ob(\mathbf{X})}\) of maps \(c_X:\mathcal{M}/X\rightarrow\mathcal{M}/X\) (where \(\mathcal{M}/X\) is a partially ordered class of isomorphism classes of monomorphisms in \(\mathcal{M}\) with codomain \(X\)) such that for every X-object \(X\), the following three conditions hold:

The above notion of categorical closure operator allows one to extend the concept of connected topological space to an object of a suitable category (see, e.g., [M. M. Clementino and W. Tholen, Topology Appl. 75, No. 2, 143–181 (1997; Zbl 0906.18003); Appl. Categ. Struct. 9, No. 6, 539–556 (2001; Zbl 0993.18004); J. Šlapal, Appl. Categ. Struct. 17, No. 6, 603–612 (2009; Zbl 1184.54014)]). The present paper follows suit and presents yet another approach to connectedness via closure operators. In particular, it defines a suitable class \(\mathcal{M}\) of monomorphisms, which is not necessarily a part of a factorization structure on a category \(\mathbf{X}\). More precisely, the paper calls a class \(\mathcal{M}\) of monomorphisms in X a domain provided that:

The paper is well written, provides most of its required preliminaries, and could be of interest to all those researchers, who study categorical topology.

- (1)
- (Extension) \(m\leqslant c_X(m)\) for every \(m\in\mathcal{M}/X\);
- (2)
- (Monotonicity) if \(m\leqslant m^{\prime}\), then \(c_X(m)\leqslant c_X(m^{\prime})\) for every \(m,\,m^{\prime}\in\mathcal{M}/X\); and, finally,
- (3)
- (Continuity) \(c_X(f^{-1}(m))\leqslant f^{-1}(c_Y(m))\) for every \(f:X\rightarrow Y\) and every \(m\in\mathcal{M}/Y\) (in which \(f^{-1}(-):\mathcal{M}/Y\rightarrow\mathcal{M}/X\) is the inverse image map given on an element \(m\in\mathcal{M}/Y\) by a pullback along the morphism \(f:X\rightarrow Y\) in question).

The above notion of categorical closure operator allows one to extend the concept of connected topological space to an object of a suitable category (see, e.g., [M. M. Clementino and W. Tholen, Topology Appl. 75, No. 2, 143–181 (1997; Zbl 0906.18003); Appl. Categ. Struct. 9, No. 6, 539–556 (2001; Zbl 0993.18004); J. Šlapal, Appl. Categ. Struct. 17, No. 6, 603–612 (2009; Zbl 1184.54014)]). The present paper follows suit and presents yet another approach to connectedness via closure operators. In particular, it defines a suitable class \(\mathcal{M}\) of monomorphisms, which is not necessarily a part of a factorization structure on a category \(\mathbf{X}\). More precisely, the paper calls a class \(\mathcal{M}\) of monomorphisms in X a domain provided that:

- (1)
- \(\mathcal{M}\) contains all the identities;
- (2)
- \(\mathcal{M}\) is stable under pullbacks, i.e., for every \(m\in\mathcal{M}/X\) and every \(f:Y\rightarrow X\) in X, a pullback \(f^{-1}(m)\) of \(m\) along \(f\) exists and lies in \(\mathcal{M}/Y\);
- (3)
- for every X-object \(X\), \(\mathcal{M}/X\) is closed under binary meets (where the meet of \(a\), \(b\) is the diagonal of a pullback of \(a\) along \(b\));
- (4)
- for every X-object \(X\), \(\mathcal{M}/X\) has the smallest element.

The paper is well written, provides most of its required preliminaries, and could be of interest to all those researchers, who study categorical topology.

Reviewer: Sergejs Solovjovs (Praha) (MR4125953)

### MSC:

18A20 | Epimorphisms, monomorphisms, special classes of morphisms, null morphisms |

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

18B99 | Special categories |

18C40 | Structured objects in a category (group objects, etc.) |

18F60 | Categories of topological spaces and continuous mappings |

54D05 | Connected and locally connected spaces (general aspects) |

### Keywords:

closure operator; connected topological space; domain; factorization structure; fine epimorphism; monomorphism; preorder; product of objects; pseudo-complement; pullback; quasi-complement; strict initial object
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\textit{A. R. Shir Ali Nasab} and \textit{S. N. Hosseini}, Bull. Iran. Math. Soc. 46, No. 4, 1195--1210 (2020; Zbl 1446.18002)

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### References:

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[13] | Slapal, J., Another approach to connectedness with respect to a closure operator, Appl. Categ. Struct., 17, 603-612 (2009) · Zbl 1184.54014 |

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