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Plus constructions are at the nexus of relative bimodules, indexedenrichments, modules over functors and graphical calculus. We define several ofthese in different settings. The first takes a category to a monoidal categoryand the second is an endofunctor for monoidal categories. There are localizedand unital versions of these constructions. These serve three main purposes.The first is to define a notion that generalizes modules over algebras tomonoidal functors and modules. This is realized via indexed enrichment. Thesecond is to provide a theory of twists, and the third is to classify relativebimodules over a given groupoid. In this guise they classify bi-module monoidswith respect to a plethysm products over a the homomorphisms of a categorythought of as a bi-module of the underlying groupoid. The presented plusconstructions generalize the plus construction for Feynman categories andexplain the appearance of the plethysm monoid definition for operads, properadsand props. To this end we introduce a new notion of unique factorizationcategory (UFC) and show that the plus construction of a hereditary UFC is aFeynman category. Just as finite sets are the fundamental indexing Feynmancategory cospans are the fundamental indexing category hereditary UFC. We givea local presentation of these constructions as well as a global description ofthe morphisms and a graphical version using decorated groupoid colored graphs.We furthermore consider an enriched setting. The global presentation utilizespasting diagram from 2-categories or equivalently double categories, which isof independent interest. The graphical description is a consequence of thelocal definition. In the special case of a UFC there is also a formalism withgroupoid colored graphs. In both cases levels appear when one adds units, whichclarifies their role in this type of construction.

Plus constructions, plethysm, and unique factorization categories with applications to graphs and operad-like theories