Unbiased multicategory theory

TL;DR

This paper develops an unbiased, categorical framework for symmetric multicategories using double categories, generalizing to various types of multicategories and connecting to monad structures.

Contribution

It introduces a novel unbiased categorical approach to symmetric multicategories via double categories and extends the framework to various generalizations.

Findings

Provides a new categorical definition of symmetric multicategories.

Shows how to generalize to infinitary and plain multicategories.

Connects multicategories with monad structures for cartesian cases.

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 $\dM$.