Abstract

We gather some not-very-well-known remarks on units in monoidal categories, motivated by (but independent of) higher-dimensional viewpoints. All arguments are elementary, some of them of a certain beauty. The first theme is uniqueness of units: we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and show that it is contractible (if nonempty). The second theme is a redundancy in the classical definition of units, which is exhibited with clarity by comparison with an alternative definition of unit originally due to Saavedra: a Saavedra unit is a cancellable idempotent, in a certain sense. It is shown that the two notions are isomorphic in a strong functorial sense. One corollary of this comparison is that a (strong) semi-monoidal functor is compatible with the left constraint if and only if it is compatible with the right constraint, and in fact this compatibility can be measured on I alone. The unit compatibility condition for a (strong) monoidal functor is shown to be precisely the condition for the functor to lift to the categories of units. The notion of Saavedra unit leads naturally to the equivalent non-algebraic notion of fair monoidal category (treated elsewhere), where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered.

Citations