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Note on total categories

Published online by Cambridge University Press:  17 April 2009

Walter Tholen
Affiliation:
Fernuniversität, Postfach 940, D-5800 Hagen, Federal Republic of Germany.
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Abstract

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It is shown that, for a semi-topological functor T: A → X, the category A is total, that is, the Yoneda embedding of A has a left adjoint, if X is total. In particular, monadic categories over Set (possibly without rank) are total, and full reflective subcategories of total categories are total.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Adámek, Jiří, “Colimits of algebras revisited”, Bull. Austral. Math. Soc. 17 (1977), 433450.CrossRefGoogle Scholar
[2]Börger, R., Tholen, W., Wischnewsky, M.B., Wolff, H., “Compactness and hypercompleteness”, preprint.Google Scholar
[3]Isbell, John R., “Small subcategories and completeness”, Math. Systems Theory 2 (1968), 2750.CrossRefGoogle Scholar
[4]Rattray, B.A., “Adjoints to functors from categories of algebras”, Comm. Algebra 3 (1975), 563569.CrossRefGoogle Scholar
[5]Street, R., Tholen, W., Wischnewsky, M.B., Wolff, H., “Semitopological functors III: lifting of monads and adjoint functors”, J. Pure Appl. Algebra (to appear).Google Scholar
[6]Street, Ross and Walters, Robert, “Yoneda structures on 2-categories”, J. Algebra 50 (1978), 350379.CrossRefGoogle Scholar
[7]Tholen, Walter, “Semi-topological functors I”, J. Pure Appl. Algebra 15 (1979), 5373.CrossRefGoogle Scholar
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