Abstract
In this chapter we provide the basic categorical framework on subobjects, inverse images and image factorization as needed throughout the book.
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Finding an adequate notion of factorization system in a category has been a theme in category theory almost from the very beginnings. Early references include Mac Lane [1948] and Isbell [1957], but it was not before the late sixties to early seventies that a generally accepted definition emerged, most comprehensively presented by Freyd and Kelly [1972], but see also Kennison [1968], Herrlich [1968], Ringel [1970], Pumplün [1972], Dyckhoff [1972] and Bousfield [1977]; it is the self-dual notion of (ε, M) factorization system as presented here in 1.8. We have chosen to take a “one-sided” approach to it via right M- factorizations (going back to Ehrbar and Wyler [1968], [1987], Tholen [1979], [1983] and MacDonald and Tholen [1982]) since idempotent closure operators “are” exactly such factorization systems, as will be made precise in 5.3. The notion of finite M-completeness and its characterization by Theorem 1.6 does not seem to have appeared previously in the literature. Theorem 1.7 goes back to Im and Kelly [1986].
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© 1995 Springer Science+Business Media Dordrecht
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Dikranjan, D., Tholen, W. (1995). Preliminaries on Subobjects, Images, and Inverse Images. In: Categorical Structure of Closure Operators. Mathematics and Its Applications, vol 346. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8400-5_1
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DOI: https://doi.org/10.1007/978-94-015-8400-5_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4631-4
Online ISBN: 978-94-015-8400-5
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