Preliminaries on Subobjects, Images, and Inverse Images

  • D. Dikranjan
  • W. Tholen
Part of the Mathematics and Its Applications book series (MAIA, volume 346)


In this chapter we provide the basic categorical framework on subobjects, inverse images and image factorization as needed throughout the book.


Commutative Diagram Inverse Image Closure Operator Natural Transformation Categorical Property 
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  1. 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].Google Scholar

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© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • D. Dikranjan
    • 1
  • W. Tholen
    • 2
  1. 1.Department of MathematicsUniversity of UdineUdineItaly
  2. 2.Department of Mathematics and StatisticsYork UniversityNorth YorkCanada

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