Abstract
Let Xbe a category with a given(E,M)-factorization structure for morphisms, M ⫆ Mono X. In general, an arbitrary endofunctor T of X fails badly to preserve the E-class. If T carries a monad structure, then T(E) ⫆ E implies that the corresponding category of Eilenberg–Moore-algebras admits (E,M)-factorizations and vice versa. In order to get T as close as possible to this nice algebraic behaviour, a couniversal modification T↪ T with (E) ⫆ E is constructed in two different ways using mild and natural assumptions on E and M, respectively. T inherits its monad structure from T. In case of T = U F, F ⊣ U, the Eilenberg–Moore-category of T contains a universal (E,M-algebraic hull (completion) of U [2, 3]. There are further applications to varietal hulls [4] and to function spaces.
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References
Adámek, J., Herrlich, H. and Strecker, G. E.: Abstract and Concrete Categories, Wiley, 1990.
Bargenda, H.: Algebraische Hüllen rechtsadjungierter Funktoren, Thesis, Universität Bremen, 1987.
Bargenda, H.: Universal algebraic completions of right adjoint functors, in Category Theory at Work, Heldermann, Berlin, 1991, pp. 289–306.
Bargenda, H. and Richter, G.: Varietal hulls of functors, Quaestiones Math. 4(1980), 121–158.
Binz, E.: Continuous Convergence on C.X/, Lecture Notes in Math. 469, Springer-Verlag, Berlin, Heidelberg, New York, 1975.
Brown, R.: Ten topologies for X _ Y, Quart. J. Math. Oxford (2) 14(1963), 303–319.
Činčura, J.: Tensor products in the category of topological spaces, Comment.Math. Univ. Carol. 20(1979), 431–445.
Eilenberg, S. and Moore, J.C.: Adjoint functors and triples, Illinois J. Math. 9(1965), 381–398.
Herrlich, H.: Regular categories and regular functors, Canad. J. Math. 26(1974), 709–720.
Herrlich, H.: Cartesian closed topological categories, Math. Colloq. Univ. Cape Town 9(1974), 1–16.
Herrlich, H.: Categorical topology 1971–1981, in Proc. 5th Prague Top. Symp., Prague 1981, Heldermann, Berlin, 1982, pp. 279–383.
Isbell, J.: A note on complete closure algebras, Math. Systems Theory 3(1969), 310–312.
Katěetov, M.: Measures in fully normal spaces, Fund. Math. 38(1951), 73–84.
Knight, C. J., Moran, W. and Pym, J. S.: The topologies of separate continuity. I, Proc. Camb. Phil. Soc. 68(1970), 663–671.
Lawson, J. and Madison, B.: On congruences and cones, Math. Z. 120(1971), 18–24.
Manes, E. G.: Algebraic Theories, Springer-Verlag, Berlin, Heidelberg, New York, 1976.
Richter, G.: Kategorielle Algebra, Akademie-Verlag, Berlin, 1979.
Richter, G.: Algebra_ Topology?!, in Category Theory at Work, Heldermann, Berlin, 1991, pp. 261–273.
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Richter, G. Coreflectivity of E-Monads and Algebraic Hulls. Applied Categorical Structures 8, 161–173 (2000). https://doi.org/10.1023/A:1008606215120
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DOI: https://doi.org/10.1023/A:1008606215120