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created coherence law
(was surprised to find that we didn’t have this already. Or do we?)
a stub for associativity, supposed to be an entry that eventually serves to unify in a coherent way (hah!) the various sub-entries on the topic that we already have
We do have coherence theorem, and coherence theorem for monoidal categories. Another reasonable page to have (maybe a synonym for coherence law?) might be coherence condition.
I think I would need a lot more convincing before accepting the thesis that all coherence laws ultimately boil down to contractibility conditions. There may be some kernel of truth to that, but the page comes off as a bit doctrinaire in my opinion (although I’d have to think hard before making concrete proposals for changes).
We do have coherence theorem,
Right, so the way I put the links currently I am saying that a coherence theorem is the assertion that a given definition does satisfy the relevant coherence law .
I find that more logical. Discussing a coherence theorem presupposes really that one has an idea of what a coherence law would be, so that one can then prove that it is satisfied.
Ah, I wasn’t aware of that one. Have linked to it now from coherence law.
Another reasonable page to have (maybe a synonym for coherence law?) might be coherence condition.
Okay, I made that a redirect
I think I would need a lot more convincing before accepting the thesis that all coherence laws ultimately boil down to contractibility conditions. There may be some kernel of truth to that, but the page comes off as a bit doctrinaire in my opinion (although I’d have to think hard before making concrete proposals for changes).
Hm, okay. My impression was that this is the obvuious nice condition. Of course (see the discussion about cohomology that we are having here every second month) the term “coherence” is not copy-righted, so everybody is free to call things coherence laws that don’t fit this bill. But those of which we are confident that they are good do seem to fit this. Well, clearly saying “the space of choices is contractible” is not a precise formal statement. So that should be made more precise.
What should we do? You want to put a caveat into the entry?
Todd, I just reminded myself of what you had written at coherence theorem for monoidal categories.
There yourself you emphasize that MacLane’s coherence theorem asserts exactly that a certain groupoid – the one that is precisely the one that deserves to be called the “groupoid of possible choices of associators” – is contractible.
To me that seems to be the statement that gets to the crux of the matter.
Yeah, I know Urs, but there are so many instances of coherence conditions out there that it would make me uncomfortable to baldly assert they all fit in this pattern. I think I said there’s a kernel of truth there, but I’d really have to think hard to be convinced that all cases boil down to this.
I left you a “query” box at coherence law. I strongly disagree with this formulation of coherence theorem.
I am much more comfortable saying that a demand for contractibility conditions is often a useful heuristic for discovering coherence laws (and in hindsight might be used to justify the choice of coherence conditions in many classical structures, although I think this might depend on what has been proven or what one believes to be true about the periodic table).
but I’d really have to think hard to be convinced that all cases boil down to this.
Okay, I currently have the opposite problem: I have to think hard to find a case that does not boil down to this!
Help me, which case are you thinking of?
Oh, I see it in the query box now.
Hm , I am not sure if I get your point. Sorry, please bear with me. This is important.
You are saying in that query box that braided monoidal categories are a counterexample to my claim, or at least make it questionable.
So where do you see a non-contractible space of choices here?
Let me ask: are you thinking of the fact that the double braiding
$B_{y \otimes x}\circ B_{x \otimes y} : x \otimes y \to x \otimes y$may not equal to the identity? In that case, just to clarify: this is not the space of choices that I mean. This is a $\pi_3$ in a one-object-one-morphism 3-category. If I’d demand that to vanish I’d be demanding every $n$-category to be the point!
What I have in mind are the choices for the structure morphisms that relax the usual definition of a category: associativity and uniticity. It would seem to me that the two additional axioms in a braided monoidal category that demand the associator to be compatible with the braiding is exactly the condition that this space of ways of using associators to relax associativity in a one-object-one-morphisms 3-category is contractible.
Of course, the query box is specifically addressing the formulation of “coherence theorem”, not “coherence law”. Over time, the meaning of “coherence theorem” has transformed a bit, and the article coherence theorem gives some idea of the scope.
I currently have the opposite problem: I have to think hard to find a case that does not boil down to this!
That’s a fair question. Please take care to note that I am still open to being convinced myself, but I think the statements being put down at coherence law need a bit of qualification or warning or something.
The thing is, you can have coherence conditions that come in from several directions but wind up interacting in complicated ways. A good idea might be symmetric monoidal closed categories. It’s sort of arguable that if you think of a symmetric monoidal category as a special type of 4-category, then the coherence laws for a symmetric monoidal category are ultimately derivable from contractibility conditions. And, there are coherence laws (usually called triangular equations) which govern adjunctions, so that the homs right adjoint to tensors are also governed by some sort of contractibility. But these coherence laws mix in very subtle ways, so that for example there are two definable maps of the form
$[[[x, I], I], I] \to [[[x, I], I], I]$in an smc category, and no one in their right mind would demand them to be equal in the theory of smc cats. Same thing happens in *-autonomous categories, and really all over the place.
Okay, that helps. So I was thinking exclusively in terms of definitions of plain $n$-categories, not $n$-categories with extra operations on them.
In that case, i think that contractibility is important, so I would like to see if we can agree in that case. Then I would tend to want to make in the entries a distinction between “coherence laws in the definition of plain n-categories” and “coherence laws for other kinds of algebraic structures”.
So what I am saying is that in cases where there is a mix of structures, so that the result is not just some case of $(n, r)$-category, the precise selection of coherence laws (and particularly where to stop imposing more) is not a cut-and-dried matter to be decided by some omnibus statement about contractibility.
If you are restricting the meaning of coherence law just to the case of $(n, r)$-categories (where $n = \infty$ is allowed), then I think I agree with you.
It’s sort of arguable that if you think of a symmetric monoidal category as a special type of 4-category, then the coherence laws for a symmetric monoidal category are ultimately derivable from contractibility conditions.
Oh, really? That surprises me. (Of course I have no explicit control over either set of coherence laws in this case. )
But let me ask directly: you are the one person on the planet who has written out coherence for 4-categories. You believe that this does not reproduce the coherence law for symmetric monoidal categories?
Wouldn’t that be shocking? That would mean that if I start with your definition of tetracategory, I can find an another sensible definition of symmetric monoidal categories which is inequivalent to the standard one. No?
Sorry, I must have misrepresented myself when I said “it’s sort of arguable”. I didn’t mean it’s debatable, I meant one could argue that… (and IMO argue successfully). In other words, we are in agreement on that. Sorry for the confusion; it was my fault.
If you are restricting the meaning of coherence law just to the case of $(n, r)$-categories (where $n = \infty$ is allowed),
Yes, that’s what i was thinking of exclusively. I think Mike also noticed this problem with me before. I tend to ignore categorical structures that are not special cases of $(n,r)$-categories. ;)
then I think I agree with you.
All right, that’s a relief! No fun finding myself in such stark disagreement with Todd Trimle on technical matters!
On the other hand, I was now already preparing my weapons and would next have started to prove to you that various n-categories obtained from accepted models of $(\infty,n)$-categories provably have the contractibility property. For instance the condition on a quasi-category $C$ is equivalent to demanding that the map $[\Delta[2],C] \to [\Lambda^1[2],C]$ has as fibers contractible $\infty$-groupoids. That’s the coherence law for associativity in the corresponding algebraic $A_\infty$-category.
Okay, then I think I will now write the entries such as to make this distinction between the two kinds of coherence conditions clear. Then you can check if you like that better.
“It’s arguable” didn’t mean it’s debatable, I meant one could argue that
Oop, sorry for misunderstanding that. Now that made me check my local disctionary. I had not, to be frank, been aware that this word can in fact be used in precisely two opposite senses.
Just for completeness, I should reply to this here, too:
But these coherence laws mix in very subtle ways, so that for example there are two definable maps of the form
$[[[x, I], I], I] \to [[[x, I], I], I]$in an smc category, and no one in their right mind would demand them to be equal in the theory of smc cats.
Yes, and the reason one wouldn’t claim this is, analogous to the example one dim down with the double braiding, that this morphism witnesses a nontrivial $\pi_4$ in a 4-category, and is not constructed exclusively from structure morphisms of the associator for 3-morphisms and its compatibilities with the 4-morphisms. Right?
@Urs #18: yes, I think you can put it that way.
It’s best to see this coming from geometric definitions of higher categories:
say we start with an $(2,1)$-category $C$ given as a simplicial set and want to build a bicategory out of it, with associators satisfying coherence.
So we choose a composition operation by choosing 2-horn fillers. Then we choose associators by filling the spheres $\partial\Delta[3] \to C$ all whose faces are made from the chosen compositions. Now, of course in general $C$ will contain non-contractible such spheres. But by the defining properties of $C$, those spheres whose four faces are compositions can be filled. So we do have associators. And then further, again by the definition properties all 3-spheres whose faces are the thus chosen associators can be filled. That’s what after we have made all our choices of composites and associators appears then to us as the coherence conditon that happens to be satisfied by these associators.
So the thing is that of course $C$ is allowed to have non-contractible higher spheres. But we are guaranteed that whatever choices for composition, associators, pentagonators etc. we make, we will never wrap these cycles.
By the way, I expanded associator.
Yes, I agree of course, except for “it’s best to see this coming from geometric definitions of higher categories”. The same philosophy holds in algebraic approaches. :-)
For instance the condition on a quasi-category $C$ is equivalent to demanding that the map $[\Delta[2],C] \to [\Lambda^1[2],C]$ has as fibers contractible $\infty$-groupoids.
In fact I would tend to say that it is not contractibility, but that certain maps are trivial fibrations, with contractible being replaced by $blah \to \ast$ a trivial fibration. cf the definition of a Segal category.
Yes, but that’s the same statement: the fibers of a trivial fibration are contractible, so that’s a parameterized collection of contractible spaces.
Specifically, in our example the fact that $[\Delta[2],C] \to [\Lambda^1[2],C]$ is a trivial fibration and hence has contractible fibers asserts all the components of the associativity coherence law between all tuples of objects.
the fibers of a trivial fibration are contractible
no. They are weakly equivalent to the point, so that any two points in a fibre are connected by a path, any two such paths are homotopic (in the path) etc (sorry for teaching you to suck eggs). I don’t know if we actually need the space to _be_contractible, even though it often is.
And saying something is a trivial fibration is neater than saying a bunch of fibres are contractible, which doesn’t tell us much about the map itself.
the fibers of a trivial fibration are contractible
no. They are weakly equivalent to the point
That’s the same!
Maybe you have in mind a model of the situation where the oo-groupoids are modled by things that are not both fibrant and cofibrant. Then it’s different. But we don’t need to do that.
ah, ok then.
Right, so maybe my use of “space” interchangebly with “$\infty$-groupoid” was misleading. In as far as “space” here is read as “topological space” it is implicitly to be assumed to be a CW-complex (the realization of some Kan complex, specifically). So Whitehead’s theorem applies and tells us that every such that is weakly equivalent to the point has a contraction. And conversely of course.
Yes - I was thinking while offline that Trimble fundamental n-groupoids genuinely have a space of choices, which is not necessarily ’nice’. Even though such choices are given, this is pretty much an arbitrary choice. One could consider a geometrisation of his algebraic definition, much as we now have an algebraisation of the geometric definition of an n-groupoid as a certain Kan complex, and then I would prefer to rely on a formulation saying ’such and such is an acyclic fibration’. I would like to see (and yes, the onus is on me to put it in) reference to complete segal spaces or the like in discussion of coherence conditions like this.
Not quite following what is trying to be said in #28.
0-cells in $\Pi_n(X)$ are points of $X$.
1-cells are paths $I \to X$.
…
$(n-1)$-cells are maps $D^{n-1} \to X$.
$n$-cells are homotopy-rel-boundary classes of maps $D^n \to X$.
There is a canonical set of operations on these cells which specifies the structure of $\Pi_n(X)$, defined by pure abstract nonsense. So where are the ’arbitrary’ choices in this specification?
I’d think the coherence law in the definition of Trimble n-category is exhibited by the fact that the topological spaces $Hom(I, I^{\vee n})$ are contractible. These are the “spaces of choices”.
Yes. Could someone explain how these choices are ’arbitrary’, or what that means precisely?
Might “arbitrariness”, where it enters in making geometric notions of higher category algebraic, be closely related to “bias”, in the sense of choosing some operations over other available ones as primitives in the theory? (I think of Trimble n-cats as an “unbiased” notion.)
Since it is a very good example, I think, I added the case of Trimble $n$-categories as one brief paragraph to the Examples-section at coherence law.
the fact that the topological spaces $Hom(I, I^{\vee n})$ are contractible
this did occur to me, and I realised that it negated my argument - I was thinking of iterated path spaces of the given space $X$, which obviously can be a bit nasty for general $X$ (but then the fundamental $n$-groupoid might not be the right construction, and some sort of profinite/prodiscrete/shape thing might be better). As far as arbitrary choices go, I may not be recalling the definition of Trimble category correctly, but Todd is correct in his appraisal of my #28 - there is a canonical set of operations, but these aren’t the only ones, if one is choosing between biased and unbiased. In my thesis when I was dealing with the definition of $\Pi_2(X)$ for $X$ a topological groupoid, one had to be very explicit and careful as to the choices of the structure maps, which essentially came from choosing elements of the various hom-spaces $C(I^n,I^m)$. In my setting I couldn’t use the canonical choices, and had to use a mix of biased, unbiased and non-standard, piecewise affine transformations. In fact my space of available choices was only a subspace of $C(I^n,I^m)$, and not one that was obviously contractible, although I’m guess it is, probably some sort of deformation retract of the full hom-space.
Hi David,
you know this, but just for amplification and for the record: the point of these operadic definitions of higher categories is that no specific choices for composites are enforced, but that instead all possible ways to compose are kept tract of while keeping them under algebraic control.
So to some extent these definitions might actually be re4garded as sitting half-way in between the purely geometric ones where just the existence of a space of composites is assserted and the bi-tri-tetracategory ones, where a speicifc composite is chosen.
Here for Trimble n-category there are just the full spaces of composites without a choice of composite as in the geometric apporach, but then on top of this is an algebraic structure that governs how these spaces of choices connect together.
added to the Examples-section at coherence law a discussion of coherence in $n$-groupoids modeled as Kan complexes
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