Elementary remarks on units in monoidal categories

TL;DR

This paper investigates an alternative, more economical definition of units in monoidal categories based on Saavedra's cancellative idempotents, showing their equivalence to traditional units and exploring implications for functoriality and higher category theory.

Contribution

It introduces and proves the equivalence of Saavedra units with traditional units, and connects this to fair monoidal categories and functorial lifting, offering a simplified perspective.

Findings

Saavedra units are contractible if nonempty

Equivalence of Saavedra and traditional units is strongly functorial

Unit compatibility condition aligns with lifting functors to units

Abstract

We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of left-right constraints, and is motivated by higher category theory. To start, we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and observe that it is contractible (if nonempty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair monoidal category, 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. The paper is self-contained. All arguments are elementary, some of them of a certain beauty.