Proper Categories and Their Duals

摘要

Let C be a category with subobjects in which every inclusion splits and every morphism has factorization. A cone y in category C is certain map from vC to C and a cone y in C is a proper cone if there is at least one component of y an epimorphism. Here we introduce the semigroup PC of proper cones in a category C and its dual P* C. Further it is also shown that the categories of left [right] ideals L(S) [R(S)] of a semigroup S with appropriate morphisms are proper categories. The semigroup PIL(S) [PIR(S)] of proper cones and their duals in these categories are also discussed.