Natural multitransformations of multifunctors

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 classical universal algebra. We introduce the notion of a natural multitransformation of multifunctors, owing to which categories of multifunctors from a multicategory to some other 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 commacategories with properties similar to those in 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 commutative FSet-operads coincide with commutative algebraic theories. © Allerton Press, Inc., 2011.