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The aim of this paper is the study of semantics, compositionality and net properties (as quasi-liveness of transitions, boundedness of places, termination of nets, deadlocks, coverness) of algebraic high-level nets in a categorical framework. We show that the algebraic high-level nets can be composed in a elegant manner using colimits and especially pushouts. We define two kinds of semantics for algebraic high-level nets: the standard semantics and normed scheme semantics, prove that both semantics are compositional, and study in which way the above net properties can be analyzed.