Cloven operadic categories: An approach to operadic categories with cardinalities in finite unordered sets

We introduce and study operadic categories with cardinalities in finite sets and establish conditions under which their associated theories of operads and algebras are equivalent to the standard framework introduced in 2015 by Batanin and Markl. Our approach is particularly natural in applications to the operadic category of graphs and the related category of modular operads and their clones. 28 pages