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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:
(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).
Given a closure operator \(C\) in a category X, an element \(m\in\mathcal{M}/X\) is called \(C\)-closed (resp. \(C\)-dense) provided that \(c_X(m)=m\) (resp. \(c_X(m)=1_X\), where \(1_X\) stands for the identity morphism on \(X\)).
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 definition of connectedness given in the paper is based in the idea that a topological space \(X\) is connected provided that there do not exist two non-empty closed subsets \(S_1\) and \(S_2\) of \(X\) with the property that \(S_1\cap S_2 =\varnothing\) and \(X=S_1\cup S_2\) (or equivalently \(X\backslash(S_1\cup S_2)=\varnothing\)). The authors then illustrate their notion with several examples and generalize two well-known topological results: on connectedness of union and product of connected topological spaces.
The paper is well written, provides most of its required preliminaries, and could be of interest to all those researchers, who study categorical topology.

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)
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References:

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