Geodesic
Abstract initiality
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 3, pp. 575-583.

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We study morphisms that are initial w.r.t. all functors in a given conglomerate. Several results and counterexamples are obtained concerning the relation of such properties to different notions of subobject. E.g., strong monomorphisms are initial w.r.t. all faithful adjoint functors, but not necessarily w.r.t. all faithful monomorphism-preserving functors; morphisms that are initial w.r.t. all faithful monomorphism-preserving functors are monomorphisms, but need not be extremal; and (under weak additional conditions) a morphism is initial w.r.t. all faithful functors that map extremal monomorphisms to monomorphisms iff it is an extremal monomorphism.
Classification : 18A10, 18A20, 18A22, 18B30
Keywords: initial morphism; (extremal) monomorphism; faithful functor; semicategory 
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Schröder, Lutz; Herrlich, Horst. Abstract initiality. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 3, pp. 575-583. http://geodesic.mathdoc.fr/item/CMUC_2000__41_3_a12/