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How complete are categories of algebras?

Published online by Cambridge University Press:  17 April 2009

Jiří Adámek
Jiří Adámek
Affiliation:
Faculty of Electrical Engineering, FEL CVUT Suchbátarova 2, Praha 6, Czechoslovakia
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Completeness properties of (i) the category Alg(T) of T-algebras over a functor T: X → X and (ii) the subcategory XT in the case where T = (T, μ, η) is a monad, are investigated. It is known that if X is compact, then each XT is compact; we present a functor T: Set → Set such that Alg(T) is non-compact, although it is hypercomplete. If T either preserves epis or has a rank, we prove that Alg(T) and XT are topologically algebraic over X provided X satisfies mild additional hypotheses. Nevertheless, a natural monad over the category of Δ-comp1ete posets is exhibited such that its category of algebras is solid, but not topologically algebraic, over Set.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 36 , Issue 3 , December 1987 , pp. 389 - 409
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Adámek, J., “Colimits of algebras revisited”, Bull. Austral. Math. Soc. 17 (1977), 433–450.CrossRefGoogle Scholar
[2]Adámek, J., Theory of mathematical structures, (D. Reidel Publ. Comp., Dordrecht-Boston-Lancaster, 1983).Google Scholar
[3]Adámek, J. and Koubek, V., “Are colimits of algebras easy to construct?”, J. Algebra 66 (1980), 226–250.CrossRefGoogle Scholar
[4]Barr, M., “Coequalizers and free triples”, Math. Z. 116 (1970), 307–322.CrossRefGoogle Scholar
[5]Börger, R. and Tholen, W., “Remarks on topologically-algebraic functors”, Cahiers Topologie Geom. Differentielle 20 (1979), 155–177.Google Scholar
[6]Börger, R., Tholen, W., Wischnewsky, M.B. and Wolff, H., “Compact and hypercomplete categories”, J. Pure Appl. Algebra 21 (1981), 129–144.CrossRefGoogle Scholar
[7]Herrlich, H., Nakagawa, R., Strecker, G.E. and Titcomb, T., “Equivalence of semi-topological and topologically-algebraic functors”, Canad. J. Math. 32 (1980), 34–39.CrossRefGoogle Scholar
[8]Hoffman, R.-E., “Semi-identifying lifts and generalization of the duality theorem for topological functors”, Math. Nachr. 74 (1976), 295–307.CrossRefGoogle Scholar
[9]Hong, S.S., “Categories in which each monosource is initial”, Kyungpook Math. J. 15 (1975), 133–139.Google Scholar
[10]Hong, Y.H., Studies on categories of universal topological algebras, (Thesis, McMaster University, 1974).Google Scholar
[11]Isbell, J.R., “Small subcategories and completeness”, Math. Systems Theory 2 (1968), 27–50.CrossRefGoogle Scholar
[12]Kelly, M., “A unified treatement of transfinite constructions for free algebras, free monoids, colimits associated sheaves, and so on”, Bull. Austral. Math. Soc. 22 (1980), 1–84.CrossRefGoogle Scholar
[13]Koubek, V., Reitermann, J., “Categorical constructions of free algebras, colimits and completions of partial algebras”, J. Pure Appl. Algebra 14 (1979), 195–231.CrossRefGoogle Scholar
[14]Manes, E.G., Algebraic Theories, (Springer, New York-Heidelberg-Berlin, 1976).CrossRefGoogle Scholar
[15]Rattray, B.A., “Adjoints to functors from categories of algebras”, Comm. Algebra 3 (1975), 563–569.CrossRefGoogle Scholar
[16]Tholen, W., “Semi-topological functors I”, J. Pure Appl. Algebra 15 (1979), 53–73.CrossRefGoogle Scholar
[17]Trnková, V., “Automata in categories”, Lect. Notes Comp. Sci. 32 (Springer, 1975).Google Scholar
[18]Trnková, V., Adámek, J., Koubek, V. and Reitermann, J., “Free algebras, input processes and free monads”, Comment. Math. Univ. Carolin. 16 (1975), 339–351.Google Scholar