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The \({\mathcal {HS}} = {\mathcal {SH}}\) problem for coalgebras. (English) Zbl 1220.03011
It has been shown by H. P. Gumm and T. Schröder [Algebra Univers. 53, No. 2–3, 229–252 (2005; Zbl 1086.08002)] that, given a set functor \(T\), if \(T\) preserves preimages then \({\mathcal {HS}}(K)={\mathcal {SH}}(K)\) for every class \(K\) of \(T\)-coalgebras, and the converse implication holds if \(|T1|>1\) (i.e., if \(T\)-coalgebras have nontrivial behaviour – if \(|T1|=1\), then the identity map on the singleton set is a final \(T\)-coalgebra). The authors of the present paper remark that the converse implication fails if \(|T1|=1\), a counterexample being the functor \({\mathcal C}_{0,1}\) sending the empty set to itself and all other sets to a singleton, and suggest that in the general case, \({\mathcal {HS}}={\mathcal {SH}}\) might be equivalent to \(T\) preserving non-empty preimages. They prove a partial case of this conjecture, namely for \(T\) failing to preserve non-empty preimages on undistinguished points, where a point is distinguished if it is in the image of a natural transformation \({\mathcal C}_{0,1}\to T\). Further partial results are given for the remaining case (\(T\) failing to preserve non-empty preimages but preserving them on undistinguished points).

MSC:
03C05 Equational classes, universal algebra in model theory
08A30 Subalgebras, congruence relations
18A99 General theory of categories and functors
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