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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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##### References:
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