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Images in categories as reflections. (English) Zbl 0632.18001
The paper under review is based on joint work by the two authors, carried out in 1968/69 and not published until now (except for a preliminary technical report). As in the past, the image of a morphism $$f: A\to B$$ of a category C is defined with respect to a subclass M of C via a factorization with a certain “global” diagonal property. This property is equivalent to the existence of a reflection for M in the category of commutative squares of C.
Although the generality of the presentation (which includes discussions of the dual concepts) may be of some interest to old-time categorists, the dependence of the notion of image on a choice of the subclass M renders the application of the results more troublesome than beneficial. Indeed, in most concrete categories the image of f is (or should be) the kernel of the pair of insertions $$B\rightrightarrows^{u}_{v}S$$ associated with the amalgamated sum S of $$B\leftarrow^{f}A\to^{f}B$$.
Reviewer: J.Sonner

##### MSC:
 18A32 Factorization systems, substructures, quotient structures, congruences, amalgams 18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) 18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
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##### References:
 [1] 1, F. Cagliari & S. Mantovani , Factorizations in topological categories and related topics , Preprint 1986 , MR 958734 · Zbl 0615.54011 [2] 2 H. , Ehrbar , Bilder und adjungierte Funktoren , Dissertation, Univ, München 1969 , 3, H. Ehrbar & O. Wyler , On subobjects and images in categories , Technical Report 68-34, Dept, Math., Carnegie-Mellon Univ . 1968 , 4, H. Ehrbar & O. Wyler , On subobjects and images in categories , Preprint 1969 , P.J. Freyd & G.M. Kelly , 5 Categories of continuous functors, I, J, Pure App , Algebra 2 ( 1972 ), 169 - 191 , MR 322004 | Zbl 0257.18005 · Zbl 0257.18005 · doi:10.1016/0022-4049(72)90001-1 [3] 6 A. Grothendieck , Sur quelques points d’algèbre homologique , Tohoku Math, J , 9 ( 1957 ), 119 - 221 , Article | MR 102537 | Zbl 0118.26104 · Zbl 0118.26104 · minidml.mathdoc.fr [4] 7, H. Herrlich , Perfect subcategories and factorizations , Colloquia Math, Soc , Janos Bolyai 8 ( 1972 ), 387 - 403 , MR 362193 | Zbl 0335.54011 · Zbl 0335.54011 [5] 8 H. Herrlich , G. Salicrup & G.E. Strecker , Factorizations, denseness, separation, and relatively compact objects , Preprint 1986 , MR 911689 · Zbl 0629.18003 [6] 9, J.R. Isbell , Subobjects, adequacy, completeness, and categories of algebras , Rozprawy Mat , 36 ( 1964 ). MR 163939 | Zbl 0133.26703 · Zbl 0133.26703 [7] 10, M. Jurchescu & A. Lascu , Morfisme stricte, categorii cantoriene, functori de completare , Studii Cerc. Math. 18 ( 1966 ), 219 - 234 , MR 220793 | Zbl 0192.34102 · Zbl 0192.34102 [8] 11 G.M. Kelly , Monomorphisms, epimorphisms, and pull-backs , J, Austral, Math, Soc , 9 ( 1969 ), 124 - 142 , MR 240161 | Zbl 0169.32604 · Zbl 0169.32604 · doi:10.1017/S1446788700005693 [9] 12, J.F. Kennison , Full reflective subcategories and generalized covering spaces, III , J, Math , 12 ( 1968 ), 353 - 365 , MR 227247 | Zbl 0155.31402 · Zbl 0155.31402 [10] 13, J. Macdonald & W. Tholen , Decomposition of morphisms into infinitely many factors , Lecture Notes in Math , 962 , Springer ( 1982 ), 175 - 189 , MR 682955 | Zbl 0497.18006 · Zbl 0497.18006 [11] 14, S. Maclane , Duality for groups , Bul. Amer. Math. Soc. 50 ( 1950 ), 485 - 516 , Article | MR 49192 | Zbl 0045.29905 · Zbl 0045.29905 · doi:10.1090/S0002-9904-1950-09427-0 · minidml.mathdoc.fr [12] 15, J. Sonner , Canonical categories , Proc, Conf, Categorical Algebra La Jolla 1965 , Springer ( 1966 ), 272 - 294 , MR 220794 | Zbl 0185.04002 · Zbl 0185.04002 [13] 16, G.E. Strecker , Perfect sources , Lecture Notes in Math , 540 , Springer ( 1976 ), 605 - 624 , MR 451192 | Zbl 0338.54007 · Zbl 0338.54007 [14] 17, W. Tholen , Factorizations, localizations and the orthogonal subcategory problem , Math. Nachr. 114 ( 1983 ), 63 - 85 , MR 745048 | Zbl 0553.18003 · Zbl 0553.18003 · doi:10.1002/mana.19831140105 [15] 18, O. Wyler , Weakly exact categories , Archiv d, Math , ( Basel ) 17 ( 1966 ), 9 - 19 , MR 190211 | Zbl 0163.01503 · Zbl 0163.01503 · doi:10.1007/BF01900199
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