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I am reading the paper
Before introducing the coherent nerve, the paper motivates the business of homotopy coherences in low dimension. It goes on with saying that the combinatorics of these becomes increasingly hard to handle in higher dimensions and that the business with coherent nerve solves this in one step. However, it seems that the combinatorics is exactly the combinatorics of Street’s orientals, but in the coherent homotopy literature nobody does a clear comparison.
Thus, I would like to understand how to get the equivalent handling using the Street’s $\omega$-nerve, and seeingt the proof that such an approach to coherent homotopy is equivalent to the one via Vogt-Cordiert-Porter machinery. I mean I would like to have precise mathematical statements, not just intutitive understanding.
Tim ? Urs ?
If I remember right, the combinatorics is different from Ross’s because, for instance, in the Vogt coherence for the 4 simplex, the model for paths from 0 to 4 is a cube hence has 6 faces. The corresponding model in the orientals would have 5. This is because the combinatorics in the Vogt model does not take interchange as an identity.
To go for that example there is (in an ad hoc notation) a face of the cube, $S[4](0,4)$, which is (012)(234), i.e. the product of two 1-simplices, giving a square face of the result. The other faces of the cube correspond to actual faces of the 4-simplex, such as (0123)(34), (0124), etc. The extra face corresponds to an interchange square.
Looking at the face diagram the oriental diagram gives the pentagon whilst the ’VCT’ approach yields a hexagon.
In higher dimensions the gap between things gets bigger as $S[5](0,5)$ is a 4-cube so has 8 faces and the 5 simplex has 6 faces. There are two interchangers: (0123)(345) and (012)(2345).
Of course, any partition of 0 to n gives a similar situation in $S[n](0,n)$.
I am not 100% sure on how to interpret all this in other than fairly simple terms, but to me this corresponds to the non-vanishing of Whitehead products in arbitrary homotopy types (or in this case perhaps in directed homotopy types of some sort.) That idea is, I think, very related to the results of
D. Blanc, M.W. Johnson and J.M. Turner, Higher homotopy operations and cohomology, J. K-Theory 5 (2010), pp. 167–200.
(Edit: I have glanced over at oriental and if I understand correctly the orientals are strict $\omega$-categories, whilst the $S[n]$ are simplicially enriched categories, and this allows non-trivial Whitehead product type constructions to survive. I think that is the point.)
I need some time to digest your answer. So you are pessimistic as of the idea that the Street’s orientals could be used at all to give an equivalent treatment of coherent homotopy ? I mean regarding that orientals are good for doing cocycles of various kind, could we look at homotopies of homotopies etc. in coherent homotopy theory as cocycles and then find the appropriate way of forming resolutions to simulate this ? I should think much more about this.
Street’s oriental are good for doing strict things (that is in their original form) so can only do a form of coherent homotopy with interchanges. (I don’t know what that would come out to be. Perhaps Urs would have an idea.)
Cordier’s resolution which gives the h.c. models for simplices IS a resolution and the cocycles would be homotopies of homotopies etc. Baues and Blanc recently looked at a cohomology of categories, but I am not 100% certain that this was among them, but I think it was. Dwyer, Kan and Smith used exactly this sort of resolution in their work and that is mentioned in Blanc, Johnson and Turner’s paper, which is well worth a glance.
So I am not pessimistic that this is one good way forward, but it depends where you want to get to. :-)
I think Tim is quite right. The orientals are the canonical cosimplicial strict $\infty$-category, whereas $\mathfrak{C}$ is a cosimplicial $(\infty,1)$-category.
But here is a point worth noticing for the direction where Zoran is coming from (as far as I know): the reason why strict $\infty$-groupoids and hence orientals go quite a long way in many applications in higher topos theory, is that the homotopy hyothesis changes character as one passes to $\infty$-sheaves:
by this cryptic sentence I mean the following evident statement: given a locally $\infty$-connected $\infty$-topos $\mathbf{H}$ with geometric realization functor $\Pi : \mathbf{H} \to \infty Grpd$ there may be many objects in $\mathbf{H}$ that are given by sheaves with values in strict $\infty$-groupoids, that nevertheless map under $\Pi$ to $\infty$-groupoids that are far from strict.
This is, Zoran, why we could use orientals so much back then, without losing generality.
For instance in H = $Smooth \infty Grpd$ there is the object presented by the sheaf of strict 2-groupoids that is given by the strict version of the String Lie 2-group. Under $\Pi$ this maps to $Sing$ of the topological string group.
The reason is that besides the categorical homotopy groups of an object in $\mathbf{H}$, under $\Pi$ these mix with the geometric homotopy groups of the degreewise sheaves. This is where lots of topological structure comes in even if the categorical homotopy type was strict.
Nice to make that observation explicit, Urs. As an extreme case, the terminal object of a locally ∞-connected ∞-topos is a sheaf with values in strict (-2)-groupoids, but $\Pi$ takes it to the fundamental $\infty$-groupoid of the ∞-topos in question—and every weak $\infty$-groupoid occurs as the fundamental $\infty$-groupoid of some ∞-topos.
As an extreme case, the terminal object of a locally ∞-connected ∞-topos is a sheaf with values in strict (-2)-groupoids, but $\Pi$ takes it to the fundamental $\infty$-groupoid of the ∞-topos in question—and every weak $\infty$-groupoid occurs as the fundamental $\infty$-groupoid of some ∞-topos.
Very good example.
In the same vein, one low dimension up, consider topological $\infty$-groupoids in $ETop \infty Grpd$ presented by sufficiently well behaved simplicial topological spaces. Under $\Pi$ these map to their geometric realization as simplicial topological spaces. So every (sufficiently tame) homotopy type is already the image under $\Pi$ of a 0-groupoid object in $ETop \infty Grpd$: the categorically discrete topolgical $\infty$-groupoid on that space.
This is maybe in some way the crucial example to ponder: it shows that we may switch the perspective on a toplogical space between
presenting a topologically discrete categorically non-discrete $\infty$-groupoid;
or presenting a topologically non-discrete but categorically discrete $\infty$-groupooid.
This is maybe in some way the crucial example to ponder
Well, if you think little as I like to, then I think my example is the crucial one to ponder, for exactly the same reason! (-:
Right, agreed. I should have known that you would say this. And you are right.
Concerning thinking big, though, I still have to sort out the following bit in order to be able to replace in the above statement the condition “suitably well behaved” with “degreewise locally contractible”:
I know that Artin-Mazur says that for $X$ locally contractible, forming a hypercover that is degreewise a coproduct of contractibles and then contracting these to the point yields a Kan complex $K$ whose homotopy groups are that of $X$.
Do they also show the slightly stronger statement that there is a weak homotopy equivalence $|K| \to X$?
(trouble is I know their result by secondary sources which don’t mention this. I should finally dig out the original literature).
Well, it seems clear that there’s a map ${|K|}\to X$, and it’d be weird if it weren’t a whe. I can check AM when I get home.
it’d be weird if it weren’t a whe
Yes, I expect it is, but I don’t know if they prove it.
I can check AM when I get home.
Thanks, Mike, that would be very helpful! I don’t have access to it currently from where I am.
Yes, they do. Their Theorem 12.1 says that if X is locally contractible and U is a hypercovering of (the little site of) X, then the simplicial set $\Pi(U)$ is (weakly?) homotopy equivalent to the singular complex of X. Actually in the proof they construct a span of (weak?) homotopy equivalences relating $\Pi(U)$ to $S(X)$, whose vertex is the diagonal of the bisimplicial set $S(U)$. Since geometric realization takes weak homotopy equivalences of simplicial sets to homotopy equivalences of CW complexes, we get an induced map ${|\Pi(U)|} \to {|S(X)|} \to X$ which is uniquely determined up to homotopy and a weak homotopy equivalence.
Does AM refer to Artin-Mazur, Etale Homotopy (SLNM 100)?
Yes.
Thank, Mike! I’ll put that on some nLab page, when I find a second.
So then the only missing ingredient for a good setup of the cohesive $\infty$-topos $LCTop\infty Grpd$ of locally contractible topological $\infty$-groupoids is the question that David Roberts started talking about recently in another thread: how to best deal with fine-tuning the definition of a site of contractible topological spaces.
I’ll maybe create a stub-page where we can work on this.
okay, I have made a quick note at locally contractible topological infinity-groupoid with some indication of what one would want to say now, and how
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