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 Pb of pullback squares in finite sets: a symmetric multicategory is a sum preserving discrete fibration of double categories M: M → Pb. If the "loose" part of M is an opfibration we get unbiased symmetric monoidal categories.
The definition can be usefully generalized by replacing Pb with another double prop P, as an indexing base, giving P-multicategories. For instance, we can remove the finiteness condition to obtain infinitary symmetric multicategories, or enhance Pb 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 M^cart are "spans" of a tight and a loose arrow in M.
Keywords: symmetric multicategories; double categories; fibrations
2020 MSC: 18C10, 18D30, 18E05, 18M05, 18M60, 18M65, 18N10
Theory and Applications of Categories, Vol. 44, 2025, No. 28, pp 826-868.
Published 2025-09-08.
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