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Calcul des satellites et présentations des bimodules à l’aide des carrés exacts. (French) Zbl 0533.18007

This paper contains the first part of a work aimed at the presentation of the definition and calculation of satellites and derived functors in the conceptual frame of the exact squares in the sense of the first author [Ann. Sci. Math. Qué. 4, 103-125 (1980; Zbl 0495.18008)]. Concretely it contains the general theory of satellites in this framework. The authors establish differences between satellites and punctual satellites (analogous to that between Kan extensions and punctual Kan extensions) showing as examples of this concept the usual Kan extensions and the mean cotensor products of F. Borceux and G. M. Kelly [Bull. Aust. Math. Soc. 12, 49-72 (1975; Zbl 0329.18011)]. The concept of presentation of a bimodule is basic in the remaining of the paper: it extends the usual concept of presentation of a module and allows to recover the usual theories of satellites.
Reviewer: M.L.Laplaza

MSC:

18E25 Derived functors and satellites (MSC2010)
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
18G10 Resolutions; derived functors (category-theoretic aspects)
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References:

[1] 1. F. Borceux , Limites enrichies et existence de V-foncteur adjoint , Cahiers Top. et Géom. Diff. XVI - 4 ( 1975 ), 396 . Numdam | MR 498791 | Zbl 0329.18013 · Zbl 0329.18013
[2] 2. D. Bourn , Natural anadeses and catadeses , Cahiers Top. et Géom. Diff. XIV - 4 ( 1973 ), 371 . Numdam | MR 354808 | Zbl 0323.18003 · Zbl 0323.18003
[3] 3. D. Buchsbaum , Homological and commutative Algebra, in Categories and commutative algebra , C.I.M.E. III Ciclo 1971 , Cremonese Roma ( 1973 ), 13 - 39 . Zbl 0285.18009 · Zbl 0285.18009
[4] 4. A. & C. Ehresmann , Multiple functors , I-IV , Cahiers Top. et Géom. Diff. : XV - 3 ( 1974 ), 215 ; XIX - 3 ( 1978 ), 295 ; XIX - 4 , 387 ; XX - 1 ( 1979 ), 59 . Numdam
[5] 5. J.W. Gray , Formal category theory , Lectures notes in Math. 391 , Springer 1974 . MR 371990 | Zbl 0285.18006 · Zbl 0285.18006
[6] 6. R. Guitart , Remarques sur les machies et les structures , Cahiers Top. et Géom. Diff. XV - 2 ( 1974 ), 113 . Numdam | MR 384891 | Zbl 0319.18003 · Zbl 0319.18003
[7] 7. R. Guitart , Des machines aux bimodules , 1978 , in : Thèse (chap. 30 , 461 - 484 ), Université de Picardie , Amiens , Juin 1979 .
[8] 8. R. Guitart , Relations et carrés exacts , Ann. Sc. Math. Qué. IV - 2 ( 1980 ), 103 . MR 599050 | Zbl 0495.18008 · Zbl 0495.18008
[9] 9. R. Guitart & C. Lair , Critères de rigidifications... , Diagrammes 6 , Paris ( 1981 ), 17 p. MR 684534 | Zbl 0514.18005 · Zbl 0514.18005
[10] 10. R. Guitart & L. Vandenbril , Décompositions et lax-complétions , Cahiers Top. et Géom. Diff. XVIII - 4 ( 1977 ), 333 - 407 . Numdam | Zbl 0381.18012 · Zbl 0381.18012
[11] 11. G.M. Kelly , Various papers in: Lectures Notes in Math. 420 , Springer ( 1974 ).
[12] 12. S. Maclane , Homology , Springer , 1963 . MR 156879 | Zbl 0133.26502 · Zbl 0133.26502
[13] 13. B. Mitchell , Cours donné à l’Université Paris 7 en 1971 - 72 .
[14] 14. R.H. Street , Various papers in : Lectures Notes in Math. 420 , Springer ( 1974 ).
[15] 15. L. Vandenbril , Exposé 23 , Séminaire de catégories de Paris 7 ( 1979 ).
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