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Here by coherence conditions I mean those axioms imposed on associators and unities that grant that the groupoid generated by such morphisms is a poset, i.e. any two parallels morphisms in this groupoid are equal. These condition implies that such weak structures are equivalent (in a suitable sense) to strict ones.

That said, where does these conditions come from? There are some deep reason, maybe motivated by some applications, for which these conditions are required?

One possible answer just for monoidal categories is that these conditions made the monoidal product more similar to the cartesian product (requiring that for every two $n$-products in the bi-category/monoidal category there's a unique isomorphism of the groupoid generated by the associators and unities).

There something similar that can be said about bi-categories?

Thanks in advance to anyone who will answer.

Edit: After reading the Zhen Lin's comment and Berci's answer I realized that this question need more specifications.

I'm aware of the fact that these axioms allows to prove the coherence theorem, which exactly says that the groupoids generate by associators and unities is a poset (as said above). What I'm really looking for is reason for the coherence at all. As I said I'm also aware the coherence provide equivalence of bi-categories/monoidal categories to strict ones, so I guess my question should be

Why do we really need that this structures are equivalent to strict ones?

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The axioms for monoidal categories as we know them today have been chosen because they are what we need to prove the coherence theorem. The original axioms had some redundancies, but were basically also chosen so that a coherence theorem could be proven for them. In some sense, the real definition of monoidal category should be the unbiased one (see Leinster's book, Higher operads, higher categories). –  Zhen Lin Nov 30 '12 at 21:04
@ZhenLin Thanks for the comment, I completely agree with what you said, I realize my question wasn't addressing my doubt, I've edited the question, hope I've made myself more clear. –  Giorgio Mossa Dec 1 '12 at 16:17
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1 Answer

up vote 3 down vote accepted

The main point is that composition of arrows in bicategories (tensor of objects in monoidal categories) wants to be associative by nature, but it is associative only up to isomorphism. However, we should really speak about multiple composition such as $$e\otimes f\otimes g\otimes h $$ in a sense, without parenthesis! This can be made precise in more ways. And the crutial thing for this is the coherence theorem, if we start out from binary tensors: we need to apply a unique passage from any $2$ parenthesized form of a multiple composition. For example, in the definition of an internal semigroup object $s$, one would write associativity like a commuting square with top arrow $(s\otimes s)\otimes s \overset{\mu\otimes s}\to s\otimes s$ and left arrow $s\otimes (s\otimes s) \overset{s\otimes\mu}\to s\otimes s$, but $(s\otimes s)\otimes s$ and $s\otimes (s\otimes s)$ should be identified to one "$s\otimes s\otimes s$".

If we only require that $(f\otimes g)\otimes h\cong f\otimes (g\otimes h)$, then we could have many choices for the passage $(s\otimes s)\otimes s\to s\otimes (s\otimes s)$ above, and it's not clear then how an internal semigroup exactly would be meant at all.

The other thing, is that in all living example, since the given tensor is always associative "deep inside", actually there is always a canonical choice of these coherence isomorphisms, and for this the prototype example is in the monoidal category $(\mathcal{Set},\times)$: the passage $(A\times B)\times C \to A\times (B\times C)$ is given by $\langle \langle a,b\rangle,c\rangle \mapsto \langle a,\langle b,c\rangle\rangle$.

I agree Zhen Lin's comment that "in some sense, the real definition of monoidal category should be the unbiased one". It basically presents $n$-ary tensors as primitive concept (the tensor is given as an $n$-ary operation for all $n\in\Bbb N$), and the coherence isomorphisms are of the form $$e_1\otimes\ldots\otimes e_n \to e_1\otimes ..(e_i\otimes..\otimes e_j)..\otimes e_n.$$

This relies on the observation, that in the living examples the multiple tensors are always present themselves in the given monoidal category, and that the ordinary coherence isomorphisms canonically factor through these (e.g. in the case of sets the above isomorphism is the composition of $(A\times B)\times C \longleftarrow A\times B\times C \longrightarrow A\times (B\times C) $).

In unbiased setting, the coherence axiom is a requirement of commutative squares finally, and basically only requires that putting any $2$ pair of parnethesis into an $n$-fold tensor, it doesn't matter which parnethesis is "unfolded" first, where unfolding means the application of the given coherence isomorphism. In this axiomatic setting the coherence theorem becomes a simple induction.

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thanks to the answer, I realized that probably my question didn't address my doubts so I've edited it. –  Giorgio Mossa Dec 1 '12 at 16:17
I think, it (at least partly) answers your intended question as well: we want to define internal semigroups, monads, adjunctions, Morita contexts, whatsoever in a bicategory. It is not clear how to define these, if the passages like between $(s\otimes s)\otimes s \to s\otimes (s\otimes s)$ are not unique. –  Berci Dec 1 '12 at 17:50
Anyway, we don't need it: I'm also very interested in 'bicategories without coherence', I remember a paper about it, but now I can't find. Turn back later. –  Berci Dec 1 '12 at 17:54
It was this one: arxiv.org/abs/math/9804106 –  Berci Dec 3 '12 at 12:12
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