Talk:Coherence condition

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[edit] Reference?

Hello, do you have a reference for this definition of "coherence condition"? I have not seen the term used in this way before, for example to describe the monad laws or composition in ordinary categories. The only use I am familiar with is as in Mac Lane's book, chapter VII, where they are conditions that give rise to significant and profound coherence theorems. Sam Staton (talk) 13:18, 21 December 2007 (UTC)

Strangely enough, I've never found an actual definition in print, only plenty of uses without definition. If you do category theory, you are apparently supposed to pick this up by osmosis – I once, too, had the term explained to me, orally. For examples of use (without definition), just use the Google search term ["monoidal category" "coherence condition" diagram]; examples not involving monoidal categories are also not hard to find: [1], [2], [3], [4], etcetera. I spotted one actual use for describing the monad laws: [5] For reasons that are not immediately clear to me, the terminology is indeed not commonly used in defining monads, although the monad laws are clearly instances of the same concept.  --Lambiam 21:08, 21 December 2007 (UTC)

Hmm. Thank you for the references. Perhaps we should try to clarify common usage, in the article. It may be that hardly anyone thinks of the associativity laws of categories as "coherence conditions". It would be odd if, as a result of this article, people started doing so; I don't think that would be the right order of events. Sam Staton (talk) 11:44, 23 December 2007 (UTC)

I've added something to the effect to the article. Please feel free to revise this as you deem appropriate.  --Lambiam 15:19, 23 December 2007 (UTC)

Hello, I still don't think the article captures the essence of coherence results. My concerns are best summarized with a quote from John Power, "A General Coherence Result", J. Pure and Applied Algebra, 1989:

"Originally, results about certain diagrams commuting used to be seen as constituting the essence of a coherence theorem. In 1972/73, Kelly argued that, although such results may be important consequences of a coherence theorem, they no longer constitute its essence. ... Kelly had conjectured that for every 2-monad on Cat, every pseudo-algebra is equivalent, in the appropriate 2-category .. to a strict algebra. .. He argued that this is perhaps the ideal coherence result."

I'd like to improve the article, to reflect these kinds of views. I'm not sure that I'm expert enough to do so. I'm not sure how universal these views are, either. I'm not sure how to proceed... All the best, Sam Staton (talk) 19:21, 10 January 2008 (UTC)

Some further quotes from Kelly, 1974, Coherence theorems for lax algebras and for distributive laws:

• p 284. "It is reasonable – and probably very common – to use the term "coherence theorem" for a result which, having found out something about [a doctrine] D* from a knowledge of its algebras, goes back to these algebras and deduces something useful about them."
• pp 286-7. "Thus proving diagrams commutative may be a tool in establishing a coherence theorem, or even a way of stating it; but it need not be the only way of getting some grip on D*."

So, you see, I am not sure that this wikipedia article is about coherence theorems, at least in this sense. Sam Staton (talk) 19:36, 10 January 2008 (UTC)