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Diagram lemmas in semi-exact $JTK$-categories

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Diagram lemmas in semi-exact JTK-categories CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES J. V. MICHALOWICZ Diagram lemmas in semi-exact JTK-categories Cahiers de topologie et géométrie différentielle catégoriques, tome 12, no 4 (1971), p. 445-467. <> © Andrée C. Ehresmann et les auteurs, 1971, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation ( Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 445 DIAGRAM LEMMAS IN SEMI-EXACT JTK-CATEGORIES by J. V. MICHALOWICZ CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE Vol. XII, 4 1. Introduction The concept of a JTK-category was introduced in [5] to provide a procedure for dealing in categories with the non-categorical concepts of subobject, quotient object, one-to-one mapping, onto mapping, embedding and quotient map. D E F INIT IO N. A category LI is called a JTK-category if there is a class T of morphisms in Q such that T and the classes J, K, L , N of morphisms defined by satisfy the following conditions (T7) Every morphism f in (1 has a representation f = j t k which is unique up to isomorphism. We use M, E, B , S, I, R , C for the classes of monomorphisms, epimor- phisms, bimorphisms, isomorphisms, identity morphisms, retractions, core- tractions, respectively, in any category and we use lower case letters as indicators; e.g., m E M, t E T , t1 E T , etc. In the jTK-category, j is de- signed to be an abstraction of the class of embeddings in 8, K of the quo- 446 tient maps, T of the one-to-one onto mappings, L of the one

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