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universal object; universal category; smooth category; homogeneous; J'onsson class; special structure
The general theory of J'onsson-classes is generalized to strongly smooth quasiconstructs in such a way that it also allows the construction of universal categories. One example of the theory is the existence of a concrete universal category over every base category. Properties are given which are (under certain conditions) equivalent to the existence of homogeneous universal objects. Thereby, we disprove the existence of a homogeneous {\it C\/}-universal category. The notion of homogeneity is strengthened to extremal homogeneity. Extremally homogeneous universal objects, for which additionally every morphism between smaller subobjects is extendable to an endomorphism, are constructed in so called extremally smooth quasiconstructs.
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