Abstract:
We continue to develop the theory of multicategories over verbal categories. This theory includes both the usual category theory and the theory of operads, as well as a significant part of the classical universal algebra. We introduce the notion of natural multitransformations of multifunctors, owing to which categories of multifunctors from a multicategory to another one turn into multicategories. In particular, any algebraic variety over a multicategory possesses a natural structure of a multicategory. Furthermore, we construct a multicategory analog of comma-categories with properties similar to the category case. We define the notion of the center of a multicategory and show that centers of multicategories are commutative operads (introduced by us earlier) and only they. We prove that the notion of a commutative FSet-operad coincides with the notion of a commutative algebraic theory.
Keywords:verbal category, multicategory, multifunctor, natural multitransformation, comma-multicategory, algebra over multicategory, center, commutative operad, commutative algebraic theory.
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