Talk:Coherence condition

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:, , , , etcetera. I spotted one actual use for describing the monad laws: 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: 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)
 * 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*."

Quantum physics
Hi. I undid an edit by 131.215.48.236. The edit introduced a citation to an appendix of an article by Kitaev. I undid the edit because it seemed to have been merely to include the reference, with no context. Mac Lane's coherence theorem does not appear here, but is at Monoidal category. So Monoidal category might be a better place to note a connection with quantum physics (though it would be a digression there too). We should assume that the edit was not merely intended to publicize the paper, but in case it was, see WP:COI. Sam (talk) 21:12, 26 June 2008 (UTC)

"Expert" template explanation
IMHO, calling any equality of morphisms a "coherence condition" seems too general, and the category-axioms example seems odd. I was under the impression that a coherence condition is something along the lines of "given a category with some extra structure (e.g. a monoidal category), a coherence condition states that every diagram that can be formed by using only morphisms from this extra structure commutes". Functor salad (talk) 12:13, 31 December 2008 (UTC)


 * I think I agree with your comments, hence the discussion above. Sam (talk) 16:11, 1 January 2009 (UTC)

Explanation of extensive edit
As several people have mentioned on this talk page, the article as it was did not seem to capture the essence of coherence as it is understood in modern mathematics. As "Functor salad", coherence is generally a claim that every diagram that can be made out of structure morphisms commutes. Reorganized/rewrote the article to try and capture this idea, and also added a reference to Mac Lane's book dealing with these ideas. I left the examples from the previous version in the new section ``further examples". I'm not really sure they do illustrate coherence. They are really just the associativity of composition. I guess this is a kind of coherence, but it seems like a strange basic example.

One possible solution I see to incorporate this example but still have an article about categorical coherence would be to make a section on "algebraic precursors to coherence", or something like that. What I am getting at is that there is a type of coherence that appears in basic algebraic structures before any categories are introduces. For instance, the usual axiom for associativity is $$ a(bc)=(ab)c $$. But when working in an associative algebra, we ignore all brackets. Thus we are in fact using many more equalities. That all these other equalities follow from the axiom can be thought of as a type of coherence theorem, albeit a very easy one to prove. The examples which currently appears in the article is somewhere between this algebraic example and the "categories with extra structure" examples which are usually called coherence. So perhaps they have a place in the story. But I think that needs to be thought through

I am not fully satisfied with the article as is, and still regard it as a stub.

PeterWT (talk) 21:36, 16 December 2009 (UTC)

the name
coherence condition vs. coherency condition? What is the difference between the two terms? And which one is appropriate for this article? Jackzhp (talk) 23:17, 23 February 2011 (UTC)

Back to MacLane
MacLane states on p.165 of the 2nd edition of his book "Categories for the Working Mathematician": 'A coherence theorem asserts: "Every diagram commutes"; more modestly, that every diagram of a certain class commutes.' Now my interpretation: Ment are diagrams in a category; and a diagram may be seen as a graph homomorphism from a shape graph to the category. So a class of diagrams may be described as the class of all diagrams with a shape graph taken from a class of shape graphs. Easy examples are: "All squares commute" (shape graph: square), or "All triangles commute" (shape graph: triangle). Both conditions are easily verified for any category with hom-sets with at most one element.

As a coherence theorem 'asserts', a coherence condition should 'require'. Hence a coherence condition should be defined as the requirement, that every diagram of a certain class commutes. Thus the notion "coherence condition" should be a synonym to the notion "coherence axiom", with two examples on p.282 in the above book.

The above-discussed examples about category axioms seem odd, because they are understood as applied to the category itself. In the case of identity it is no coherence condition, because id is a map from the class of objects to the class of morphisms of the category, and no morphism. Similar the associativity involves the composition as a map from the class of composable pairs of morphisms to the class of morphisms of the category. Things look different, if category objects in a category are described. Then the category-axioms are applied in such a way, that morphisms are used to describe the constraints. So, regarding a category as a category object in the category Set of sets and (total) maps, the "odd" coherence conditions make a certain sense, although they are not satisfied by the category Set. If they were provably true, that could be stated as a "coherence theorem". Those coherence conditions will be satisfied for some subcategory of Set, at least for the empty category, and for the one-element monoid.

Things look difficult in the MacLane-examples, because there the involved object(-variable)s are (variables for) functors, and the morphism(-variable)s are (variables for) natural transformations, so things play in some category of functors and natural transformations. — Preceding unsigned comment added by 217.246.172.54 (talk) 16:28, 11 October 2011 (UTC)

...MacLane continued
Another example is "Braided Coherence", p.263. Note that MacLane puts "coherence" in quotes. So he probably did not regard "coherence" as a well-defined mathematical concept. It should better be taken as a kind of headline for some axioms that express a kind of matching (described by templates of commutative diagrams), comparable to coherent light in a laser beam, where it is defined like monochromatic, parallel light in the same phase. Or like "Braided Coherence" the word "coherence" has to be composed with some context to give a well-defined conceptual sense. — Preceding unsigned comment added by 217.246.183.138 (talk) 10:02, 12 October 2011 (UTC)

Reference
The reference to "Maclane pentagon is some recursive square" appears to be self-published, unvetted, and self-promoting, and doesn't belong. Antonfire (talk) 14:26, 8 September 2017 (UTC)