## Separation versus connectedness.(English)Zbl 0906.18003

For a category $${\mathcal X}$$ with finite products and a closure operator $$C$$ an object $$X$$ is $$C$$-separated ($$C$$-connected) if the diagonal $$\Delta_X$$ in $$X\times X$$ is $$C$$-closed ($$C$$-dense, respectively). Here closure operator is intended in the sense of Giuli and the reviewer [cf. E. Giuli and D. Dikranyan, Topology Appl. 27, 129-143 (1987; Zbl 0634.54008)] where these notions where introduced. While the subcategory $$\Delta(C)$$ of $$C$$-separated objects is better understood and described via the diagonal theorem [cf. E. Giuli and H. Hušek, Ann. Mat. Pura Appl., IV. Ser. 145, 337-346 (1986; Zbl 0617.54006) and E. Giuli, S. Mantovani and W. Tholen, J. Pure Appl. Algebra 51, No. 1/2, 129-140 (1988; Zbl 0651.18002)] the subcategory $$\nabla(C)$$ of $$C$$-connected objects was studied only recently in the monograph [D. Dikranyan and W. Tholen, “Categorical structure of closure operators. With applications to topology, algebra and discrete mathematics”, Mathematics and its Applications 346, Kluwer Academic Publishers, Dordrecht (1995; Zbl 0853.18002)].
The present authors obtain a stronger form of the diagonal theorem and introduce the important notion of coregular closure operator that permits to describe similarly the subcategory $$\nabla(C)$$ of $$C$$-connected objects. The pairs disconnectedness/connectedness are considered often as a pair of left constant and right constant categories. The authors contribute essentially also in this direction by proving that the category $$\nabla(C)$$ is left constant iff $$C$$ is regular, while the category $$\Delta(C)$$ is right constant iff $$C$$ is coregular. The authors give a very elegant diagram (Union Jack) of four commutative triangles of Galois correspondences involving closure operators of a category $${\mathcal X}$$, subcategories of $${\mathcal X}$$ and morphisms of $${\mathcal X}$$. Only one of these triangles was known previously: namely the factorization of the Pumplün-Röhrl connection obtained by G. Castellini, J. Koslowski and G. E. Strecker [Topology Appl. 44, No. 1-3, 69-76 (1992; Zbl 0774.18002)] for idempotent closure operators and by Tholen and the reviewer [loc. cit.] in the general case. The paper contains also many nice examples.

### MSC:

 18B30 Categories of topological spaces and continuous mappings (MSC2010) 54B30 Categorical methods in general topology 54D05 Connected and locally connected spaces (general aspects) 18E40 Torsion theories, radicals
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### References:

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