added to the Examples-section at coherence law a discussion of coherence in $n$-groupoids modeled as Kan complexes

]]>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.

]]>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.

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.

]]>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.)

]]>Yes. Could someone explain how these choices are ’arbitrary’, or what that means precisely?

]]>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”.

]]>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?

]]>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.

]]>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.

]]>ah, ok then.

]]>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.

]]>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.

]]>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.

]]>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, 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. :-)

]]>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.

]]>@Urs #18: yes, I think you can put it that way.

]]>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?

]]>“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.

]]>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.

]]>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.

]]>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?

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.

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