Abstract
The continuous maps p : E → B for which p* : Top/B → Top/E reflects isomorphisms are shown to coincide with the universal quotient maps as characterized by Day and Kelly. Monadicity of p* turns out to be a local property. This is used to prove the main result of the paper, namely that p* is monadic for every locally sectionable map p : E → B. There are therefore important classes of maps p for which spaces over B are equivalently described as spaces over E which come equipped with a simple algebraic structure: local homeomorphisms, locally trivial quotient maps, surjective covering maps, etc. Finally, the monadic decomposition of p* is examined for arbitrary maps p.
Partial financial support by NSERC (Canada) and by CNR (Italy) for both authors while working on this paper is gratefully acknowledged.
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References
J. Adámek, H. Herrlich and J. Rosický, Essentially equational categories, Cahiers Topologie Géom. Différentielle Catégoriques 29(1988) 175–192.
J. Adámek, H. Herrlich and W. Tholen, Monadic Decompositions, J. Pure Appl. Algebra 59(1989) 111–123.
A. Arhangel'skii, Bicompact sets and the topology of spaces, Dokl. Akad. Nauk SSSR 153(1963) 743–746, English translation: Soviet Math. Dokl. 4(1963) 1726–1729.
R. Börger and W. Tholen, Strong, regular, and dense generators, preprint (York University, Toronto 1989).
B.J. Day and G.M. Kelly, On topological quotient maps preserved by pullbacks or products, Proc. Cambridge Phil. Soc. 67(1970) 553–558.
T. tom Dieck, Transformation Groups, de Gruyter (Berlin, New York 1987).
I.M. James, Fibrewise Topology, Cambridge University Press (Cambridge, New York 1989).
G. Janelidze, Precategories and Galois theory, preprint (Milan 1990).
G. Janelidze and W. Tholen, Pull-back functors, in preparation.
A. Joyal and M. Tierney, An extension of the Galois Theory of Grothendieck, Memoirs Amer. Math. Soc. 309(1984).
J.L. MacDonald and A. Stone, The tower and regular decomposition, Cahiers Topologie Géom. Différentielle Catégoriques 23(1982) 197–213.
S. MacLane, Categories for the Working Mathematician, Springer-Verlag (Berlin, New York 1971).
E. Michael, Bi-quotient maps and cartesian products of quotient maps, Ann. Inst. Fourier 18(1968) 287–302.
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Dedicated to Max Kelly on the occasion of his sixtieth birthday.
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© 1991 Springer-Verlag
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Janelidze, G., Tholen, W. (1991). How algebraic is the change-of-base functor?. In: Carboni, A., Pedicchio, M.C., Rosolini, G. (eds) Category Theory. Lecture Notes in Mathematics, vol 1488. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084219
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DOI: https://doi.org/10.1007/BFb0084219
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