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My question is about monoidal categories. To motivate it, let me first recall something about group objects.

Assume you define a group object in a category $C$ with products by an object $G$ together with morphisms $G \times G \to G, G \to G, * \to G$, so that the diagrams commute which correspond to the group axioms. Then you want to generalize some elementary properties from usual groups ($C=Set$) to these group objects, but it is quite hard to write down the relevant diagrams. For example when you want to prove that there is a actually at most one unit $* \to G$, a morphism $G \to G'$ between group objects which respects the multiplication already respects the inversion and the unit, left-inverse implies right-inverse, etc. But all this may be reduced to the case $C=Set$ by using the Yoneda-Lemma and the the definition of a group object as an object together with a factoriation of its hom-functor over the category of usual groups. Then these calculations are easy and you don't have to produce all these diagrams.

My question is: Is there a similar definition for a monoidal category? Specifically, I want to see a neat (diagram-free?) proof of Lemma 3.2.5 in this note (Pareigis' lectures on quantum groups), which is intuitive and does not come up with diagrams without motivating them. (For me, "We tried to prove it and finally this diagram worked" is no intuition.) The lemma implies for example that the endomorphism monoid of the unit object of a monoidal category is abelian, which is quite surprising (for me). This Lemma is just one example. It seems to me that monoidal categories are "weak monoids in the 2-category Cat", but I don't see yet if this description actually simplifies these diagrams.

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Is there an author or title known for the note that you linked above? – Scott Carter Nov 28 '10 at 17:37
Todd Trimble’s answer below is an excellent answer to your general question. But for the specific case of seeing why the endo-monoid of a unit object is abelian, this is the Eckman-Hilton argument, and can be presented in ways which are (I think) extremely intuitive: see e.g. about a third of the way through – Peter LeFanu Lumsdaine Nov 28 '10 at 19:51
Dear Peter, this argument works in strict monoidal categories, right? Basically my problems arise when all these identification compatibilities come into play. – Martin Brandenburg Nov 28 '10 at 20:33
up vote 8 down vote accepted

The kind of thing you are looking for applies not just to monoidal categories, but to bicategories, and it is called the bicategorical Yoneda lemma. If $B$ is a small bicategory, one may form the strict 2-category $[B^{op}, Cat]$ consisting of weak 2-functors (aka homomorphisms), pseudonatural transformations, and modifications from $B^{op}$ to $Cat$. Then there is a Yoneda embedding

$$y: B \to [B^{op}, Cat]$$

sending an object $b$ to $\hom(-, b)$, and the mapping of $B$ onto its image is a bi-equivalence. The image however is a strict 2-category, and hence we get from this a coherence theorem which assures us that all definable diagrams (in say the free monoidal category generated by a discrete category) commute. This line of thinking was introduced by Street in his paper Fibrations in Bicategories (for which there is also a correction).

This is really the modern point of view on coherence in monoidal categories. An exposition which focuses on monoidal categories can be found (I believe) in Braided Tensor Categories by Joyal and Street.

It is true that monoidal categories can be described as weak monoids, but this by itself doesn't solve coherence issues.

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Is it possible to deduce that $\lambda(I)=\rho(I) : I \to I \otimes I$ from an equivalence to a strict monoidal category? – Martin Brandenburg Nov 28 '10 at 21:25
Todd, I'm a great fan of the Yoneda lemma, but does this really help with proving Lemma 3.2.5 as in the question? Surely you'd already need Lemma 3.2.5 (or its bicategorical analogue) to prove the Yoneda lemma. In fact on the face of it you need it to construct the Yoneda map in the first place. I do agree of course that the Yoneda lemma is very useful; it could, for example, deal with Martin's object to Peter's comment about the Eckmann-Hilton argument. – Steve Lack Nov 28 '10 at 21:29
Martin, in answer to your question about $\lambda(I)=\rho(I)$: yes. In fact the original definition of monoidal category given by Mac Lane included this as well as the two parts of Lemma 3.2.5 quoted above as part of the definition. The fact that they follow from the other axioms was proved by Kelly. – Steve Lack Nov 28 '10 at 21:31
Steve: yes, that's right -- one needs to lay some groundwork to get the Yoneda lemma off the ground, so this response is really directed to the question which begins the third paragraph in Martin's question. As you know, these lemmas were originally proven by Kelly in 1964 (J. Alg. 1, 397-402). I actually have a private method for proving these types of things from scratch, and I'm trying to make up my mind whether to write it up here (it's the type of thing not so easy to write though). Perhaps I'll write it at some point on my web at the nLab and then direct Martin's attention to it. – Todd Trimble Nov 28 '10 at 23:42
Todd, please do write up your method. I think that, if it's sufficiently advertised, then many people would cite it and use it. I know I would. – Theo Johnson-Freyd Nov 28 '10 at 23:56

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