Geodesic
Natural multitransformations of multifunctors
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2011), pp. 58-71

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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, comma-multicategory, algebra over multicategory, center, commutative operad, commutative algebraic theory.
Mots-clés : natural multitransformation 
@article{IVM_2011_11_a6,
     author = {S. N. Tronin},
     title = {Natural multitransformations of multifunctors},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {58--71},
     publisher = {mathdoc},
     number = {11},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2011_11_a6/}
}
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S. N. Tronin. Natural multitransformations of multifunctors. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2011), pp. 58-71. http://geodesic.mathdoc.fr/item/IVM_2011_11_a6/