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# Elementary remarks on units in monoidal categories

By Joachim Kock
*Math. Proc. Cambridge Phil. Soc.*,
**144** (2008), 53-76.

#### 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.

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Last updated: 2008-07-13 by
Joachim Kock.