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A monoidal category has (among other things) a pair of natural transformations called the left and right unitors

$$ \lambda_A:I\otimes A \cong A $$ $$ \rho_A:A\otimes I \cong A $$

The categories I work with on a daily basis have $\lambda_I=\rho_I:I\otimes I \cong I$, but I think that is mostly a consequence of the fact that most of them have a terminal object as their $I$. Is this true in general?

I'm sort of hoping this is true, because otherwise it seems that some of the definitions of enriched categories need to make an arbitrary choice of one unitor or the other. For example, see display (1.10) of Max Kelly's introduction to enriched category theory on page 10.

(I'm actually interested in the even-more-general case of premonoidal categories, but mathematicians don't use those very often, so I've posed the question in terms of monoidal categories instead).

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  • $\begingroup$ Yes. This is always true. I think it is in Joyal and Street's paper "Braided Monoidal Categories". $\endgroup$
    – Chris Schommer-Pries
    Commented Aug 10, 2010 at 1:01
  • $\begingroup$ It's also on page 159 of MacLane's Categories for the Working Mathematician as one of three diagrams involving the associator and the two unitors that must commute (the other two being the usual pentagon and triangle diagrams) that define a monoidal category. $\endgroup$
    – Alicia Garcia-Raboso
    Commented Aug 10, 2010 at 1:33
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    $\begingroup$ It turns out that MacLane is pretty much the only reference that requires it as an axiom. In ncatlab.org/nlab/show/monoidal+category Todd Trimble says that it was proven to follow from the other axioms in Max Kelly's 1964 paper On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc. (Journal of Algebra 1, 397–402) $\endgroup$
    – Alicia Garcia-Raboso
    Commented Aug 10, 2010 at 1:50
  • $\begingroup$ @Chris and Alberto: you should leave your comments as answers, as they certainly answer the question. $\endgroup$
    – Theo Johnson-Freyd
    Commented Aug 10, 2010 at 2:25
  • $\begingroup$ Thanks for the Mac Lane reference! It's on page 163 in my version (was there more than one edition?). Here's the DOI for the Kelly paper: dx.doi.org/10.1016/0021-8693(64)90018-3 $\endgroup$
    – Adam
    Commented Aug 10, 2010 at 2:37

2 Answers 2

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In Categories for the Working Mathematician MacLane included $\lambda_I = \rho_I$ as one of three diagrams involving the associator and the two unitors that must commute (the other two being the usual pentagon and triangle diagrams) as the axioms defining monoidal categories.

It was however proven to follow from the other axioms in Max Kelly's 1964 paper On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc. (Journal of Algebra 1, 397–402)

EDIT: For the case of premonoidal categories, I think $\lambda_I = \rho_I$ follows from the results of section 4 of John Power's Premonoidal categories as categories with algebraic structure (Theoretical Computer Science 278, 1-2, 303-321). In that paper, the unitors in a premonoidal category are defined to be central natural transformations; further down, the centre of a premonoidal category is defined to be the category with the same objects but with only the central morphisms of the original category, and this turns out to be a monoidal category, hence reducing to Kelly's proof.

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  • $\begingroup$ Argh, hrm, now I'm not so sure anymore. The theorem in question in the Kelly paper is Theorem 6, but it relies on either (5) or its dual (7) which assume $\lambda_{B\otimes C}=(\lambda_B\otimes \text{id}_C)\circ\alpha_{I,B,C}$ -- but that's not one of the Mac Lane axioms. Hrm. $\endgroup$
    – Adam
    Commented Aug 10, 2010 at 3:02
  • $\begingroup$ Actually what you want is Kelly's theorem 3', which combines his theorems 6 (which is what you quoted) and 7 (which says that (1) and (6), Mac Lane's other axioms, imply (5) and (7)). $\endgroup$
    – Mike Shulman
    Commented Aug 10, 2010 at 3:13
  • $\begingroup$ I would be very interested to hear whether Kelly's proof goes through for premonoidal categories; I don't know them well enough myself to tell. $\endgroup$
    – Mike Shulman
    Commented Aug 10, 2010 at 3:18
  • $\begingroup$ Ah, I see, Theorem 7 gets you from the (now-standard) pentagon and triangle to the extra condition of (5). I'll need to spend a bit more time with this, but it looks like I'm back on track. $\endgroup$
    – Adam
    Commented Aug 11, 2010 at 14:06
  • $\begingroup$ PS, regarding Edit, I'm specifically investigating lifting the assumption that unitors and associators are central. I think Powers' choice there was sufficient but not quite necessary to get good behavior. $\endgroup$
    – Adam
    Commented Aug 26, 2010 at 21:51
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The coherence conditions for a monoidal category assure that if an isomorphism of two expressions follows from the built-in natural isomorphisms, then it is the unique such isomorphism. In particular, there is a unique isomorphism $I\otimes I \to I$ that can be constructed only from the unitors and the associator. (There may be plenty of other isomorphisms $I\otimes I \to I$ in your particular monoidal category; there is a unique one in the free monoidal category.)

This is a general philosophy in n-category theory. A collection of "coherence" axioms are "good" if they imply that the space of choices is contractible. Recall that an $n$-category is contractible if it is nonempty, it is an $n$-groupoid, and for each object (it suffices to check at any particular object) the endomorphisms of that object are a contractible $(n-1)$-category. I.e. a contractible $n$-category is one that's $n$-equivalent to $\{\rm pt\}$. The theory of monoidal categories is an example of a theory with "good" coherence axioms: the space of isomorphisms that follow from the monoidal structure between any two objects is contractible. Other examples of "good" theories include the usual theory of (only weakly associative) 2-categories, and Lurie's theory of $(\infty,1)$-categories.

Unfortunately, I don't know what a "premonoidal category" is, so I don't know if its coherence axioms are good in the above sense.

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  • $\begingroup$ A premonoidal category gives you (for each object $A$) a pair of functors $f\mapsto \text{id}_A\otimes f$ and $f\mapsto f\otimes \text{id}_A$ instead of a bifunctor $f,g\mapsto f\otimes g$. It's a more primitive starting point for constructing the hierarchy that works up to cartesian closed categories; there are some interesting stopping points along the way. $\endgroup$
    – Adam
    Commented Aug 10, 2010 at 2:43
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    $\begingroup$ Note that in some situations, one does want a collection of coherence axioms which do not imply the space of choices is contractible. Probably the most common example is a braided monoidal category; distributive categories are another. $\endgroup$
    – Mike Shulman
    Commented Aug 10, 2010 at 3:17

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