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Terminal coalgebras in well-founded set theory. (English) Zbl 0779.18004
Summary: This paper shows that, in order to obtain the theorem of Aczel and Mendler on the existence of terminal coalgebras for an endofunctor on the category of sets, it is entirely unnecessary to delve into such exotica as non-well-founded set theory. In addition, we discuss the canonical map from the initial algebra for an endofunctor on sets to the terminal coalgebra and show that in many cases it embeds the former as a dense subset of the latter in a certain natural topology. By way of example, we calculate the terminal coalgebra for various simple endofunctors.

MSC:
18B05 Categories of sets, characterizations
18A15 Foundations, relations to logic and deductive systems
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[1] Aczel, P.; Mendler, N., A final coalgebra theorem, (), 357-365
[2] Barr, M.; Wells, C.F., Triples, toposes and theories, (1985), Springer Berlin · Zbl 0567.18001
[3] Barr, M.; Wells, C.F., Category theory for computing science, (1990), Prentice-Hall Englewood Cliffs, NJ · Zbl 0714.18001
[4] Freyd, P.J., Algebraically complete categories, (1991), preprint · Zbl 0815.18005
[5] Lambek, J., Subequalizers, Canad. math. bull., 13, 337-349, (1970) · Zbl 0201.02302
[6] Makkai, M.; ParĂ©, R., Accessible categories, () · Zbl 1291.18008
[7] N.P. Mendler, P. Panangaden and R.L. Constable, Infinite objects in type theory, in: Proc. First Annual IEEE Symp. on Logic in Computer Science, 249-257.
[8] Smyth, M.B.; Plotkin, G.D., The category-theoretic solution of recursive domain equations, SIAM J. comput., 11, 761-783, (1983) · Zbl 0493.68022
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