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According to MacLane, a monoidal category is symmetric if we have maps $\gamma_{a,b}:a\Box b\cong b\Box a$ and coherence relations, one in particular which states $\gamma_{a,b}\circ\gamma_{b,a}=1$. Is this relation "evil" in the sense of category theorists? That is, should it be an isomorphism of morphisms rather than equality?

This question can just be generall extended to a question regarding commutative diagrams in general. My experience thus far has been that such diagrams show equality of maps, but is this a much stronger notion than isomorphism? Does this question make sense?


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An additional question: If the category is generated by one object, so we really only need consider $a\Box a$, is this question entirely irrelevant? That is, there is not really any way of telling the left $a$ from the right $a$ so in that case it MUST be the identity (especially if $a$ has no elements). – Jon Beardsley Jan 2 '12 at 19:40
I have the sneaking suspicion that the answers to your original two questions are: "it doesn't matter." That is, I think you can just replace $\gamma$ with some other thing where the composition is the identity, and I think you could replace any such diagram with an isomorphic one that commutes on the nose. (I'm more certain of the latter than the former.) – Dylan Wilson Jan 2 '12 at 22:46
@JBeardz: Unless you're working in a $n$-category, $n > 1$, there isn't an interesting notion of isomorphism of arrows. Equality of arrows is in general not considered evil, at least when doing ordinary category theory. (It is probably considered evil when doing higher category theory.) – Zhen Lin Jan 3 '12 at 1:23
@Zhen: I believe he means the usual notion of an isomorphism of arrows in the category $Fun(\Delta^1, \mathcal{C})$ – Dylan Wilson Jan 3 '12 at 2:08
Regarding your additional question (asked in a comment... you should really ask it in the body of the question!), there is no reason for the map $\gamma_{a,a}$ to be the identity. It is easy to construct examples (consider complexes with their usual monoidal structure, for example) – Mariano Suárez-Alvarez Jan 3 '12 at 3:58
up vote 8 down vote accepted

I found the coherence axioms for monoidal categories very confusing until I realized that what they are really trying to say is the following. Given a monoidal category $\mathcal{M}$ and a sequence $O_1, \dots, O_n$ of objects (ordered), there are various ways of making sense of $$O_1 \otimes \dots \otimes O_n$$ all of which correspond to a choice of parenthesization. If we have associator isomorphisms as in the definition of a monoidal category, then we can get functorial maps between one choice of parenthesization and another; however, one actually gets several maps corresponding to different choices of associators. So a priori the order of parenthesization matters only up to isomorphism, but the isomorphism is not uniquely determined a priori. The coherence axioms state that the isomorphism is unique (whichever choice of associators you make, you'll get the same maps); that is, $O_1 \otimes \dots \otimes O_n$ can be defined in a manner unique up to canonical isomorphism. In the symmetric monoidal case, the axiom states that $\prod_A O_a$ can be defined in a manner unique up to canonical isomorphism when $A$ is an arbitrary finite (unordered) set.

There is a more general notion than a commutative diagram in a higher category. For instance, in a 2-category (here, just a strict one), one can say that a square diagram is 2-commutative; this means that the two ways of going around a diagram are related by a natural transformation. This leads to the notion of a 2-fibered product. The notion of a 2-fibered product is important when you want to keep track of higher morphisms; for instance, it is the appropriate notion for stacks (which form a 2-category).
Another example is in topology; one can think of a 2-commutative diagram as a diagram which commutes up to a specified homotopy.

In a monoidal higher category, the coherence axioms become 2-commutative diagrams instead of plain commutative diagrams, and the 2-morphisms in the 2-commutativity themselves have to satisfy coherence conditions of their own. This is kind of messy, but Lurie has developed a way of hiding the coherence axioms of a monoidal category in DAG II (so as to generalize to $(\infty, 1)$-categories). I don't really understand enough to say too much here.

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thanks! I appreciate this a lot. Good stuff. :) – Jon Beardsley Jan 3 '12 at 19:39

I found the idea about how not to prove the following

Theorem (Coherence for monoidal categories) Every monoidal category is equivalent to some strict monoidal category.

(which comes from Leinster book, page 15) somehow illuminating. Another good reference is May's notes on TQFTs; coehrence is treated in par. 18, page 24.

Have a good reading!

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