Unbiased multicategory theory

Abstract

We present an unbiased theory of symmetric multicategories, where sequences are replaced by families. To be effective, this approach requires an explicit consideration of indexing and reindexing of objects and arrows, handled by the double category $\dPb$ of pullback squares in finite sets: a symmetric multicategory is a sum preserving discrete fibration of double categories $M: \dM\to \dPb$. If the \"loose" part of $M$ is an opfibration we get unbiased symmetric monoidal categories. The definition can be usefully generalized by replacing $\dPb$ with another double prop $\dP$, as an indexing base, giving $\dP$-multicategories. For instance, we can remove the finiteness condition to obtain infinitary symmetric multicategories, or enhance $\dPb$ by totally ordering the fibers of its loose arrows to obtain plain multicategories. We show how several concepts and properties find a natural setting in this framework. We also consider cartesian multicategories as algebras for a monad $(-)^\cart$ on $\sMlt$, where the loose arrows of $\dM^\cart$ are \"spans" of a tight and a loose arrow in $\dM$.