Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

July 18, 2008

Some ω-Questions

Posted by Urs Schreiber

I have some questions on ω\omega-categorical issues in the context of descent and cohomology to the experts. Some of them are accompanied by the figures collected here.

Question 1) Weak equivalences of orientals.

With respect to the model category structure on ωCat\omega\mathrm{Cat} an ω\omega-functor F:C wDF : C \stackrel{\simeq_w}{\to} D between ω\omega-categories is a weak equivalence if it is essentially kk-surjective for all kk \in \mathbb{N} in the sense of definition 4 of Baez-Shulman.

Let O(Δ n)O(\Delta^n) be Street’s nnth oriental, i.e. the free ω\omega-category on the parity nn-simplex. It seems to be true that O(Δ n)O(\Delta^n) is weakly equivalent to the point

n:O(Δ n) wpt. \forall n \in \mathbb{N} : O(\Delta^n) \stackrel{\simeq_w}{\to} pt \,.

Is that right? Is there a statement and a proof of this in the literature?


Question 2) ωCat\omega\mathrm{Cat}-enrichment of cosimplicial ω\omega-categories

Consider the standard Gray-like closed monoidal structure on ωCat\omega\mathrm{Cat} and consider the category ωCoSimp:=Funct(Δ,ωCat) \omega\mathrm{CoSimp} := Funct(\Delta, \omega\mathrm{Cat}) of cosimplicial ω\omega-categories. It seems to me that using the inner hom in ωCat\omega\mathrm{Cat} degreewise cosimplicial ω\omega-categories become naturally enriched over ωCat\omega\mathrm{Cat}. I am not sure yet about the best formal way to say this, but the simple idea is indicated in figure 2.

Am I right about the ωCat\omega\mathrm{Cat}-enrichment of ωCoSimp\omega\mathrm{CoSimp}? Has this been stated anywhere in the literature? What’s the best formal way to characterize it?


Question 3) descent ω\omega-categories

The above worries me slightly because it seems obviously right to me, but then the question is why Ross Street defines the descent ω\omega-category with coefficients in the cosimplicial ω\omega-category EE not simply as

hom ωCoSimp(O(Δ ()),E). hom_{\omega\mathrm{CoSimp}}(O(\Delta^{(-)}), E) \,.

This seems to be clearly the right answer. And it seems to me that the definition he actually gives (p. 339 here, p. 32 here) is related to that by using the hom-adjunction in ωCat\omega\mathrm{Cat} to move the ω\omega-glob around.

So I am thinking the following: Street’s formula for descent ω\omega-categories is actually to be thought of as being the very formalization of the ωCat\omega\mathrm{Cat}-enrichment of ωCoSimp\omega\mathrm{CoSimp} that I was asking for above.

Does that sound right?


Question 4) codescent and homotopy category of ωCat\omega\mathrm{Cat}

For YXY \to X a regular epimorphism in Spaces\mathbf{Spaces} (your favorite choice), and Y :Δ opSpacesY^\bullet : \Delta^{op} \to \mathbf{Spaces} the corresponding simplicial space, finally for A C:Spaces opωCat(Spaces)\mathbf{A}_C : \mathbf{Spaces}^{op} \to \omega\mathrm{Cat}(\mathbf{Spaces}) an ω\omega-category valued presheaf of the form A C:Xhom(Π 0(X),C) \mathbf{A}_C : X \mapsto hom(\Pi_0(X), C) for CωCat(Spaces)C \in \omega\mathrm{Cat}(\mathbf{Spaces}) a fixed ω\omega-category internal to Spaces\mathbf{Spaces} and where Π 0(X)\Pi_0(X) denotes the discrete ω\omega-category over XX, then define, following Street, the codescent ω\omega-groupoid Π 0 Y(X)\Pi_0^Y(X) by the property that

Desc(Y ,A C)hom(Π 0 Y(X),C) Desc(Y^\bullet,\mathbf{A}_C) \simeq hom(\Pi_0^Y(X), C)

for all CC.

I know the codescent ω\omega-category Π 0 Y(X)\Pi_0^Y(X) for the case that CC is restricted to be an nn-category, for low nn. For instance, for n=1n=1 we have that Π 0 Y(X)\Pi_0^Y(X) is the familiar Čech groupoid of the cover YY.For higher nn it is the Čech groupoid with composition equalities replaced by higher cells.

For the cases (low nn) that I understand explicitly, we have weak equivalences

Π 0 Y(X) wΠ 0(X). \Pi_0^Y(X) \stackrel{\simeq_w}{\to} \Pi_0(X) \,.

If the answer to question 1) is positive, then this should be true in general. Anything in the literature on that?

If I replace in the definition of CC-valued nonabelian cohomology

H(,A C):=colim Yhom(Π 0 Y(X),C) H(-, \mathbf{A}_C) := colim_Y hom(\Pi_0^Y(X), C)

the colimit on the right with something going out of all ω\omega-categories weakly equivalent to Π 0(X)\Pi_0(X) I get something very close to the formulation of nonabelian cohomology in terms of homotopy categories of simplicial presheaves as reviewed by Toën. Is anything known about this?

Posted at July 18, 2008 10:48 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1744

24 Comments & 1 Trackback

Re: Some ω-Questions

Question 6) homotopy hypothesis for generalized spaces

Forgot this one: this concerns figure 5.

In the context of the homotopy hypothesis we pick a notion of Groupoid\infty\mathbf{Groupoid} and pick a notion of Spaces\mathbf{Spaces} such that each space has an fundamental \infty-groupoid and each \infty-groupoid has a geometric realization as a space – and ask if these two operations yield some kind of equivalence GroupoidsSpaces\infty\mathbf{Groupoids} \simeq \mathbf{Spaces}.

Consider this for the choice Spaces:=Sheaves(Euclid) \mathbf{Spaces} := \mathrm{Sheaves}(\mathbf{Euclid}) where Euclid\mathbf{Euclid} is the full subcategory of Manifolds\mathbf{Manifolds} on vector spaces and the choice Groupoids:=ωGroupoid(Spaces) \infty\mathbf{Groupoids} := \omega\mathbf{Groupoid}(\mathbf{Spaces}) (ω\omega-groupoids internal to Spaces\mathbf{Spaces}).

Let Π ω:SpacesωGroupoids(Spaces) \Pi_\omega : \mathbf{Spaces} \to \omega\mathbf{Groupoids}(\mathbf{Spaces}) be the functor which sends each space XX to its fundamental ω\omega-groupoid Π ω(X)\Pi_\omega(X) whose kk-morphisms are thin homotopy classes of smooth images of the standard kk-disk in XX, (constant in a neighbourhood of the boundary of the disk).

Let ||:ωGroupoids(Spaces)Spaces |-| : \omega\mathbf{Groupoids}(\mathbf{Spaces}) \to \mathbf{Spaces} be the functor which sends each ω\omega-groupoid CC to the sheaf |C|:Uhom(Π ω(U),C). |C| : U \mapsto hom(\Pi_\omega(U),C) \,.

First sub-question: this |||-| is defined without mentioning the ω\omega-nerve of CC. It should nevertheless yield the right notion of geometric realization in Spaces\mathbf{Spaces}. Is that right?

Second subquestion: we have Quillen model category structures on both sides. Do |||-| and Π ω\Pi_\omega induce a Quillen model equivalence? If not, what else do we get? At least an adjunction one way or the other?

Posted by: Urs Schreiber on July 18, 2008 4:15 PM | Permalink | Reply to this

Re: Some ω-Questions

Hi Urs,

which notion of a $\w$-groupoid are you choosing? I suppose that in the strict case (factoring maps through thin homotopy) it may be not entirely trivial to shown that the compositions descend to the quotient.

More later,

João

Posted by: Joćo Faria Martins on July 18, 2008 4:58 PM | Permalink | Reply to this

Re: Some ω-Questions

which notion of a ω\omega-groupoid are you choosing?

Not sure which precise aspect you are asking about. So let me be fully explicit (you will know all this, but just to be fully explicit):

I use ω\omega-category in the standard sense of “strict \infty-category”, for instance p. 305 in The algebra of oriented Simplices, or section 1.4 in Tom Leinster’s book.

An ω\omega-category internal to spaces is one where all collections of kk-morphisms are objects in Spaces\mathbf{Spaces} and all structure maps (source, target, identities, compositions) are morphisms in Spaces\mathbf{Spaces}.

An ω\omega-groupoid internal to Spaces\mathbf{Spaces} for me is an ω\omega-category internal to spaces with specified strict inverses for all kk-morphisms for all kk, such that the inverse-assigning maps are morphisms in Spaces\mathbf{Spaces}.

I suppose that in the strict case (factoring maps through thin homotopy) it may be not entirely trivial to shown that the compositions descend to the quotient.

Ah, so this question is about the definition of the fundamental ω\omega-groupoid Π ω(X)\Pi_\omega(X) in my sense.

There are two issues here:

a) Why is the result of dividing out thin homotopy still in Spaces\mathbf{Spaces}? Here the answer is: because there is a trivial way in which every quotient in Spaces=Sh(Euclid)\mathbf{Spaces} = Sh(\mathbf{Euclid}) is still in there. Take parameterized path space of XX, given by the sheaf

PX:Uhom(U×I,X) P X : U \mapsto hom(U\times I, X)

then the space of paths modulo thin homotopy is the sheaf

P thinX:Uhom(U×I,X)/ thin. P_{\sim_{thin}} X : U \mapsto hom(U\times I, X)/_{\sim_{\mathrm{thin}}} \,.

In words: a smooth family of thin-homotopy classes of paths in XX is one which has a smooth family of representative parameterized paths.

b) How does composition in Π ω(X)\Pi_\omega(X) work? Here the answer is: build it inductively first in the cubical world and then restrict to the globular world.

So we start with the 1-category P 1(X)P_1(X) as defined for instance here.

We get a double-category from that by considering P 1(Mor(P 1(X)))P_1(Mor(P_1(X))). We restrict that as usual to the 2-category inside (constant vertical paths). And so on.

Posted by: Urs Schreiber on July 18, 2008 5:25 PM | Permalink | Reply to this

Re: Some ω-Questions

Hi!

Where it appears “factoring maps through thin homotopy” it should be “considering maps up to thin homotopy.”

Posted by: Joao on July 18, 2008 6:54 PM | Permalink | Reply to this

Re: Some ω-Questions

Hi João,

now I am not sure: let me know if my reply above answered your question satisfactorily.

Posted by: Urs Schreiber on July 18, 2008 7:51 PM | Permalink | Reply to this

Re: Some ω-Questions

Hi!

Sorry for the delay.


Thought about this yesterday evening (and several times before), and even though it is not immediate that the composition of $n$-paths descends to the quotient under thin homotopy, this is true, and thus there exists the fundamental $\omega$-groupoid of a smooth manifold (and probably also of any smooth space).

The issues that I believe needed discussing (and I think you did not consider this in your argument, maybe I am missing something) was that to compose two $n$-paths up to thin homotopy one needs to choose one thin $n$-path connecting then, and it is not tautological that the composition is independent of this choise.

However, looking at the contruction of Brown and Higgins of the fundamental $\omega$-groupoid of a filtered space all the construction seems to go through without any problem to this problem.

An interesting side problem would be proving that this fundamental thin $\omega$-groupoid functor from smooth spaces to $\w$-groupoids preserves colimits. Again I am convinced that all Brown-Higgins contruction can be translated to this setting.

The articles are: Brown, Ronald and Higgins, P. J;Colimit theorems for relative homotopy groups, Journal of Pure and Applied Algebra 22: 41, 1981 and “On the algebra of cubes”. Journal of Pure and Applied Algebra, 21:233–260, 1981


In the 2-dimensional case, Roger Picken and I constructed the thin categorical group of a manifold (all the construction goes through to the fundamental thin double groupoid of a smooth manifold). We used a slightly different argument, based on lemmas 50 and 51 of the article below, which even though it can’t be translated to the general case, yields two interesting exact sequences, see 1.4.6.

The article is http://arxiv.org/abs/0710.4310,
this just reminds me that we should substitute it by an up-to-date version!

Hope this is useful.

João

Posted by: Joćo Faria Martins on July 19, 2008 10:45 AM | Permalink | Reply to this

Re: Some ω-Questions

Thanks for the references! I wasn’t actually aware of your On 2-dimensional holonomy.

Just had a quick look. You define connection data for a 2-connection on the total space PP of a principal 1-bundle PXP \to X. This would translate to the usual cocycles by taking Y:=PY := P as the surjective submersion. But then for nonabelian gerbes/2-bundles the connection data is

(A,B)Ω 1(Y,Lie(G))×Ω 2(Y,Lie(E))(A,B) \in \Omega^1(Y,Lie(G))\times \Omega^2(Y,Lie(E))

(satisfying potentially fake-flatness but nothing else)

and then

(g,a)Ω 0(Y× XY,G)×Ω 1(Y× XY,Lie(E))(g,a) \in \Omega^0(Y \times_X Y,G) \times \Omega^1(Y\times_X Y, Lie(E))

satisfying a relation with the two possible pullbacks of (A,B)(A,B). For your choice Y:=PY := P the function gg is given canonically, so need not appear explicitly. But then it will satisfy the cocycle condition on Y× XY× XYY \times_X Y \times_X Y with trivial

(f)Ω 0(Y× XY× XY,E)(f) \in \Omega^0(Y \times_X Y \times_X Y, E).

But I need to have a closer look at your article.

The issues that I believe needed discussing (and I think you did not consider this in your argument, maybe I am missing something) was that to compose two nn-paths up to thin homotopy one needs to choose one thin nn-path connecting then, and it is not tautological that the composition is independent of this choise.

What I indicated in my reply was a recursive construction:

one first defines composition for thin-homotopy classes for 1-paths, where it is easy, using thin homotopies as in (2.1) here.

Then one notices that this works for 1-paths in any diffeological space, as in section 4.1 here.

Using that, one can take 2-paths in XX to be certain 1-paths in path space PXP X and apply the previous argument to see that composition works horizontally and vertically.

And so on. It seems to me that the fact that all exchange laws hold in this recursive definition follows from abstract nonsense about the relation between cubical and globular strict \infty-categories. Concretely checking it involves using thin homotopies as on p. 32 here.

Posted by: Urs Schreiber on July 19, 2008 12:23 PM | Permalink | Reply to this

Re: Some ω-Questions

Hi!

I see now (looking at http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.0663v1.pdf), you are considering a different notion of thin homotopy! With this notion, I agree with you.

I was considering the notion of Caetano-Picken-Mackaay, and possibly others, which is defined by restricting the rank of the homotopy. In this case, some discussion is needed, unless someone proves that the two notions are equivalent.

I remember that this notion (or something very similar) also appears in

http://arxiv.org/PS_cache/gr-qc/pdf/9311/9311010v2.pdf

(Representation Theory of Analytic Holonomy C* Algebras
Authors: Abhay Ashtekar, Jerzy Lewandowski)


Posted by: Joćo Faria Martins on July 19, 2008 2:36 PM | Permalink | Reply to this

Re: Some ω-Questions

Hi,

you are considering a different notion of thin homotopy!

Well, let’s see: in lemma 4.2 we observe that a map being thin is equivalent to saying that all pp-forms of certain rank pulled back along it vanish.

So for ordinary manifolds both definitions of thin homotopy coincide. But the definition in terms of pullbacks of forms has the property that it generalizes directly to diffeological spaces (and in general to sheaves on manifolds – being sheaves, it is easy to speak about forms on them but hard to speak about vector fields). That, then, makes it easy to talk about thin homotopy classes of paths in path spaces and in higher mapping spaces.

Thanks for the pointer to

Abhay Ashtekar, Jerzy Lewandowski: Representation Theory of Analytic Holonomy C *C^* Algebras

I forget if I have seen this before. (I don’t seem to list it in the history.)

Could you point me to which page the statement appears on that you are thinking of here?

Posted by: Urs Schreiber on July 19, 2008 3:39 PM | Permalink | Reply to this

Re: Some ω-Questions

Hi!

Look at page 7, the notion that the authors define is the notion of holonomic loops, and there seems to be one equivalence relation for each Lie algebra. (They consider $U(1)$ and $SU(2)$, as far as I can remember)

It is indeed a nice result that the two relations of thin homotopy are the same. Will look at your proof.

Is the rank of a smooth map not defined in a diffeological space?

Need to rush!

João

Posted by: Joćo on July 19, 2008 4:10 PM | Permalink | Reply to this

Re: Some ω-Questions

Look at page 7, the notion that the authors define is the notion of holonomic loops

Ah, now I see what you mean: if two paths are thinly homotopic, then the parallel transport along them of any connection will coincide.

I am not, however, talking about taking that as a definition of thin-homotopy here. It is a comparatively sophisticated statement!

I instead use that other definition of thin homotopy, together with the nonabelian Stokes theorem, to derive this fact that thin homotopic paths have the same parallel transport.

It is indeed a nice result that the two relations of thin homotopy are the same. Will look at your proof.

We do spell out the proof, but it is not supposed to be a deep statement: if the rank of the map is less than kk, then clearly all kk-forms pulled back along it vanish. Conversely, if all these pulled back kk-forms vanish then then rank has to be less than kk, since if it were not, we could construct a kk-form which picks up the contribution of the kk-linearly independent tangent vecors in a neighbourhood of a point where the rank of the map is kk.

Is the rank of a smooth map not defined in a diffeological space?

There are ways to make it work. But it is not natural in a precise sense:

one accesses the diffeological space by means of maps from test domains into it. That means you can handle all those differential geometric structures on it naturally which you can pull back along these maps to your test domains. For instance, by its very definition, a differential form on a diffeological space is a collection of forms for all smooth maps from test domains into the diffeological space, compatible under pullbacks in such a way that they behave as if they were really all pulled back from some manifold.

(More technically, a form on the diffeological space is a morphism of sheaves from the diffeological space itself to the deRham sheaf, which assigns to any test domain the set of forms on it.)

So the point is: sheaves are contravariant functors and forms can be pulled back. So the two go along nicely. But vectors want to be pushed forward instead. So they work nicely when we look at generalized smooth spaces which are defined by smooth maps out of them. (But even then it is not quite as nice as with forms: while vectors can be pushed forwards, push-forward of vector-fields is less immediate).

We had a bit of discussion about this general issue in the context of Andrew Stacey’s Comparative Smootheology

Need to rush!

Sure, thanks for all your comments. I’ll see now if I can print out your article with Roger Picken somewhere.

Next week the third part of my article with Konrad will come out (finally) and we also discuss Wilson surfaces for 2-bundles/nonabelian gerbes with connection. So I am feeling a little dumb for not having been aware of your work with Picken. But I’ll have a look now.

Posted by: Urs Schreiber on July 19, 2008 6:20 PM | Permalink | Reply to this

Re: Some ω-Questions

Hi Urs,

I may be missing something, but it would appear to me that the premise of your question 1 is incorrect. That is, it is certainly not the case that the unique ω\omega-functor !:𝒪 npt!:\mathcal{O}_n\rightarrow \text{pt}, from the n thn^{\text{th}} oriental to the point, is a weak equivalence in the folkloric model category structure on ωCat\omega-\text{Cat}.

By definition, as you metioned, the weak equivalences in this structure are the ω\omega-functors f:𝔻f:\mathbb{C}\rightarrow\mathbb{D} which are essentially surjective on all cells.

That is to say if xx and yy are parallel (n1)(n-1)-cells in \mathbb{C} (which means that they have common (n2)(n-2)-sources and targets) and there is an nn-cell zz in 𝔻\mathbb{D} with (n1)(n-1)-source f(x)f(x) and (n1)(n-1)-target f(y)f(y) then there exists an nn-cell ww in \mathbb{C} with (n1)(n-1)-source xx and (n1)(n-1)-target yy such that f(w)f(w) is equivalent to zz in 𝔻\mathbb{D}.

This condition, however, is certainly not satisfied by !:𝒪 npt!:\mathcal{O}_n\rightarrow\text{pt}.

To see why this is the case, start by observing that 𝒪 n\mathcal{O}_n has a unique non-identity nn-cell, which we might call α n\alpha_n, and that this cell has (n1)(n-1)-source s n1(α n)s_{n-1}(\alpha_n) and (n1)(n-1)-target t n1(α n)t_{n-1}(\alpha_n) which are distinct and parallel (n1)(n-1)-cells.

Of course, the point pt\text{pt} as an ω\omega-category consists only of a single 00-cell \bullet along with its identity cell at each dimension. So it follows that !! maps both of s n1(α n)s_{n-1}(\alpha_n) and t n1(α n)t_{n-1}(\alpha_n) to the (unique) identity on \bullet at dimension (n1)(n-1). Furthermore, it also follows that the identity on \bullet at dimension nn provides us with a n nn-cell whose (n1)(n-1)-source and (n1)(n-1)-target are both equal to !(s n1(α n))=!(t n1(α n))!(s_{n-1}(\alpha_n))=!(t_{n-1}(\alpha_n)).

So we may regard the nn-identity on \bullet as being an nn-cell with nn-source !(t n1(α n))!(t_{n-1}(\alpha_n)) and nn-target !(s n1(α n))!(s_{n-1}(\alpha_n)) (note the orientation here, identities are reversible). Now were !! a weak equivalence, we could appeal to essential surjectivity and find an nn-cell in 𝒪 n\mathcal{O}_n whose nn-source is t n1(α n)t_{n-1}(\alpha_n) and whose nn-target is s n1(α n)s_{n-1}(\alpha_n). In other words, it would allow us to find an nn-cell in 𝒪 n\mathcal{O}_n whose nn-orientation is reversed relative to that of α n\alpha_n. So since s n1(α n)s_{n-1}(\alpha_n) and t n1(α n)t_{n-1}(\alpha_n) are distinct it follows that this nn-cell cannot be equal to α n\alpha_n and cannot be an identity, thus contradicting the fact that α n\alpha_n is the unique non-identity nn-cell of 𝒪 n\mathcal{O}_n.

Indeed, if we work just a little harder more can be said. For instance, although I haven’t checked the details, it appears clear that we should be able to prove that any ω\omega-category which is weakly equivalent to an ω\omega-groupoid is itself an ω\omega-groupoid. However, in a precise sense 𝒪 n\mathcal{O}_n is about as far from being an ω\omega-groupoid as you can get. Indeed, this observation is pretty much a direct restatement of Street’s (very strong) loop freeness results for the parity complex which generates 𝒪 n\mathcal{O}_n.

Posted by: Dominic Verity on July 19, 2008 3:26 AM | Permalink | Reply to this

Re: Some ω-Questions

the premise of your question 1 is incorrect.

Right, thanks, of course the inverses are missing. I made this mistake because I am looking at a situation where eventually all O nO_ns are mapped into ω\omega-groupoids.

Please allow me to try to fix my question:

I assume just as O nO_n is the free ω\omega-category on the nn-simplex, we can also consider U nU_n (does this have an established symbol?), the free ω\omega-groupoid on the nn-simplex (having strict inverses for all kk-morphisms for all kk).

Is it right that these U nU_n are weakly equivalent to the point?

Posted by: Urs Schreiber on July 19, 2008 10:42 AM | Permalink | Reply to this

Frobenius Orientals?

I can and should probably try to improve question 1) further.

The point of my question, to emphasize this maybe, is that I think that for a correct notion of cohomology we need to ensure that the codescent ω\omega-groupoid Π 0 Y(X)\Pi_0^Y(X) which I mention in the above entry is weakly equivalent to the discrete ω\omega-category over the space XX, which I denoted Π 0(X)\Pi_0(X).

But the kk-cells in Π 0 Y(X)\Pi_0^Y(X) are built from “orientals in Y Y^\bullet” and I think that unless we restrict in the universal property defining it the ω\omega-categories CC to be ω\omega-groupoids, then Π 0 Y(X)\Pi_0^Y(X) will fail to be weakly equivalent to Π 0(X)\Pi_0(X) in as far as the orientals fail to be weakly equivalent to the point.

It seems to me that we can throw in a couple of morphisms and relations to the orientals to achieve the desired condition. The required additional stuff seems to be of “Frobenius type” in the following sense:

Observe that for CC a strict monoidal category and BC\mathbf{B}C the corresponding strict one-object 2-category we have for the degenerate space and its degenerate cover (YX):=(ptpt) (Y \to X) := (pt \to pt) that Desc(Y ,hom(Π 0(),BC))=Monoids(C). Desc( Y^\bullet, hom(\Pi_0(-),\mathbf{B}C) ) = Monoids(C) \,.

The “right” notion of descent, DescDesc', however should be such, I think, that Desc(Y ,hom(Π 0(),BC))=FrobeniusMonoids(C). Desc'( Y^\bullet, hom(\Pi_0(-),\mathbf{B}C) ) = FrobeniusMonoids(C) \,.

This should amount to adding to the definition of O(Δ 3)O(\Delta^3), for instance, all 2-cells from one edge to two edges (the reversed triangles, giving the coproduct) as well as a 3-morphism from two such reversed triangles to the two others (the co-associativity) plus a 3-morphism between mixed combinations of the original and the reversed triangles (the Frobenius property).

If moreover we add units and counits in this Frobenius sense, it should yield a 3-category weakly equivalent to the point. Hm, I need to formalize better.

(If we label the triangles with invertibles, then the difference here disappears.)

So maybe a useful question is:

Question 7)

What is the “minimal extension” of O(Δ n)O(\Delta^n) that does make it weakly equivalent to the point?

Posted by: Urs Schreiber on July 19, 2008 1:53 PM | Permalink | Reply to this

Re: Some ω-Questions

as far from being an ω\omega-groupoid as you can get.

one would expect some sort of fundamental localiser though that made ! a weak equivalence, much like Quillen A (see e.g. this article

Posted by: David Roberts on July 22, 2008 4:43 AM | Permalink | Reply to this

Re: Some ω-Questions

David wrote

one would expect some sort of fundamental localiser though that made ! a weak equivalence, much like Quillen A (see e.g. this article

Thanks. I still need to have a closer look at this article. But not tonight.

Maybe it would help me if I could make you give me some more hints about some of the relevant aspects of what your remark is hinting at. Would be much appreciated.

Posted by: Urs Schreiber on July 23, 2008 10:11 PM | Permalink | Reply to this

Re: Some ω-Questions

Apologies for the briefness. A fundamental localiser is a class of functors which acts like a class of weak equivalences. The motivating example (for me - I’m clearly not an expert on this) is functors between (small) categories whose nerve is a weak homotopy equivalence of simplicial sets. This is the content of Quillen’s Theorem’s A and B, and seized on by Grothendieck to give us the general notion.

Essentially a fundamental localiser is a class of functors WW between small categories such that

  • It contains identities, satisfies 2of 3 and retracts rr such that iri \circ r is in WW (this is known as weak saturation)
  • If the category CC has a terminal object then CptC \to pt is in WW
  • It contains functors in the slice category Cat/CCat/C such that the induced functors on the homotopy fibres over CC are in WW

(This is from section 7 of the file I linked to earlier. Note that in that paper fundamental localisers are called ‘weak equivalence classes’.)

It would be interesting to get an ω\omega-version of this, and see if there is a fundamental localiser containing the funtors O nptO_n \to pt (O nO_n the n thn^{th} oriental), or at least characterise it.

The other thing would be to consider O\mathbf{O}-sets (or O\mathbf{O}-spaces, if you like) of presheaves on the category of orientals (I presume such a thing exists).

Posted by: David Roberts on July 30, 2008 3:36 AM | Permalink | Reply to this

Re: Some ω-Questions

I thought about a concrete proposal to replace the orientals by things that are equivalent to the point in such a way that one obtains a notion of descent that agrees with that of Street for ω\omega-groupoidal coefficients but is more restrictive for general coefficients, where it induces Frobenius conditions and higher generalizations of these.

I have some tentative notes (4 pages) with a few more details.

The main point is the definition, for each set SS, of an ω\omega-category P ω(S)P_\omega(S) that behaves like the fundamental ω\omega-category of SS regarded as a discrete contractible space.

By construction, such P ω(S)P_\omega(S)s are weakly equivalent to the point. Moreover, I think by restricting to S=[n]={0,1,2,,n}S = [n] = \{0,1,2, \cdots, n\} they do arrange themselves in a cosimplicial ω\omega-category P ω:ΔωCat P_\omega : \Delta \to \omega\mathrm{Cat} and hence induce a notion of cohomology. I am not sure for n>2n \gt 2, but up to n=2n=2 I think I can show that this cohomology agrees with that obtained using orientals iff the coefficients are 2-groupoid valued.

I might be mixed up, though. In any case, the definition of P ωP_\omega which I give in the notes needs to be formalized further.

Posted by: Urs Schreiber on July 28, 2008 1:48 PM | Permalink | Reply to this

Re: Some ω-Questions

I thought about a concrete proposal […]

Ah, I overlooked that the source and target of the Frobenius move on the last page if 3 is replaced by 1 are both connected to the identity on 0120 \to 1 \to 2. This forces product and coproduct to be invertible morphisms, after all.

So proper Frobenius structures in 2d appear this way only from n=3n=3 on. Hm…

Posted by: Urs Schreiber on July 29, 2008 10:18 AM | Permalink | Reply to this

Re: Some ω-Questions

While I am thinking of such things, rather than painting ceilings which is my foreground task for this weekend, let me comment on questions 2 and 3.

First some notation, we take \otimes to denote the (lax) Gray tensor on ω\omega-Cat. This is a non-symmetric tensor which has left and right closures, which we will denote by lax l(,𝔻)\text{lax}_l(\mathbb{C},\mathbb{D}) and lax r(,𝔻)\text{lax}_r(\mathbb{C},\mathbb{D}) respectively. We adopt this notation because these are precisely the appropriate (left and right handed) generalisations of the 2-categories of 22-functors, lax-natural transformations and modifications of the 2-dimensional theory.

Now, although \otimes is not symmetric (or braided) as a monoidal structure on ω\omega-Cat, we may still do enriched category theory relative to it - so long as we take a little care with our definitions. In general, we can follow much of the elementary presentation of enriched category theory given in the first few chapters of Max Kelly’s book on this topic. Alternatively, we could think of ω\omega-Cat as a 1-object bicategory and use Street’s presentation of enrichment in a bicategory to provide us with the appropriate background.

In particular we may use the right closure lax r\text{lax}_r to enrich ω\omega-Cat over itself. Notice here that were we to use lax l\text{lax}_l for this we would not obtain a category enriched wrt \otimes, but would instead obtain a category enriched wrt the reverse tensor X Y=YXX\otimes^\circ Y = Y\otimes X.

Furthermore, Δ\Delta can also be made into a category enriched wrt \otimes, simply by regarding each of its homsets as a discrete ω\omega-category.

Now we are able to follow Kelly and enrich the functor category [Δ,ωCat][\Delta,\omega-\text{Cat}], using the right closure lax r\lax_r and certain kinds of limits in ω\omega-Cat called ends. Explicitly, if FF and GG are two (enriched) functors from Δ\Delta to ω\omega-Cat then the hom ω\omega-category between them is given by the formula

(1)[Δ,ωCat] r(F,G)= nΔlax r(F(n),G(n))[\Delta,\omega-\text{Cat}]_r(F,G) = \int_{n\in\Delta} \text{lax}_r(F(n),G(n))

wherein we use the integral notation (as introduced by MacLane) for ends.

Observe now that if \mathbb{C} is any ω\omega-category, then by definition an ω\omega-functor [Δ,ωCat](F,G)\mathbb{C}\rightarrow [\Delta,\omega-\text{Cat}](F,G) corresponds to a family of ω\omega-functors lax r(F(n),G(n))\mathbb{C}\rightarrow\text{lax}_r(F(n),G(n)) which is extra-ordinarily natural in nΔn\in\Delta (by the universal property of ends). Such a family, in turn, corresponds to a family of ω\omega-functors F(n)G(n)\mathbb{C}\otimes F(n)\rightarrow G(n) which is natural in nΔn\in\Delta (using the adjunction F(n)lax r(F(n),)-\otimes F(n)\dashv \text{lax}_r(F(n),-)). Consequently, taking \mathbb{C} to be the 1-glob 𝔾 1=𝒪(G 1)\mathbb{G}^1=\mathcal{O}(G^1), we find that the 1-cells of [Δ,ωCat](F,G)[\Delta,\omega-\text{Cat}](F,G) correspond to ω\omega-functors 𝔾 1[Δ,ωCat](F,G)\mathbb{G}^1\rightarrow[\Delta,\omega-\text{Cat}](F,G) which in turn correspond precisely to the cells pictured in Urs’ figure 2. Taking \mathbb{C} to be the nn-glob 𝔾 n1=𝒪(G n)\mathbb{G}^{n-1}=\mathcal{O}(G^n) we recover analogous descriptions of the nn-cells of this hom-ω\omega-category.

So we find that the hom-ω\omega-category defined in the end formula above is precisely the ω\omega-category described in Urs’ pictures.

However, we have now also succeeded in showing that the descent ω\omega-category of Street and Urs’ suggestion are indeed one in the same, as conjectured in question 3. What Urs’ calls hom ωCoSimp(𝒪(Δ ()),)\text{hom}_{\omega\text{CoSimp}}(\mathcal{O}(\Delta^{(-)}),\mathcal{E}) is simply the enriched homset I would denote by [Δ,ωCat] r(𝒪(Δ ()),)[\Delta,\omega-\text{Cat}]_r(\mathcal{O}(\Delta^{(-)}), \mathcal{E}), as in the end formula above. Then as we have seen, nn-cells of that hom-ω\omega-category correspond to families of ω\omega-functors 𝒪(G nΔ m)𝔾 n𝒪(Δ m) n\mathcal{O}(G^n\otimes \Delta^m)\cong\mathbb{G}^n\otimes\mathcal{O}(\Delta^m)\rightarrow \mathcal{E}^n which are natural in nΔn\in\Delta. And these in turn are simply the elements of the (un-enriched) homset [Δ,ωCat](𝒪(G nΔ ),)[\Delta,\omega-\text{Cat}](\mathcal{O}(G^n\otimes \Delta^\bullet),\mathcal{E}) given by Street in his definition of Desc()\text{Desc}(\mathcal{E}).

BTW On the topic of the Gray tensor product of strict ω\omega-categories, Street’s nerve construction allows us to give a purely simplicial construction of this tensor. For those who are simplicially minded this approach provides a simpler and more direct presentation of the Gray tensor, which can be used to circumvent problems that arise from its traditional presentation in terms of colimits of free ω\omega-categories on products of globs. More details on this topic can be found in my recently published Memoir of the AMS on the topic of Complicial Sets.

Posted by: Dominic Verity on July 19, 2008 7:27 AM | Permalink | Reply to this

Re: Some ω-Questions

So we find that the hom-ω-category defined in the end formula above is precisely the ω\omega-category described in Urs’ pictures.

Thanks a lot. This, and in particular the detailed description you give, is very helpful.

Posted by: Urs Schreiber on July 19, 2008 10:44 AM | Permalink | Reply to this

Re: Some ω-Questions

As I was thinking about some of Urs’ subsequent questions, I had reason to re-read my post on questions 2 and 3 and realised that I had committed a superscript typo which changed the meaning of what I said.

The penultimate line of the penultimate paragraph should have read:

“families of ω\omega-functors 𝒪(G nΔ m)𝔾 n𝒪(Δ m) m\mathcal{O}(G^n\otimes\Delta^m)\cong \mathbb{G}^n\otimes\mathcal{O}(\Delta^m) \rightarrow\mathcal{E}^m which are natural in mΔm\in\Delta.”

Posted by: Dominic Verity on July 20, 2008 3:30 AM | Permalink | Reply to this

Re: Some ω-Questions

Here is what I am imagining to say:

Quickly recall the basic setup and notation I am using:

Euclid\mathbf{Euclid}: the full subcategory of Manifolds\mathbf{Manifolds} on vector spaces.

Spaces:=Sh(Euclid)\mathbf{Spaces} := Sh(\mathbf{Euclid}): sheaves on Euclid\mathbf{Euclid} with respect to the standard notion of covers.

ωCat(Spaces)\omega\mathrm{Cat}(\mathbf{Spaces}): ω\omega-categories internal to Spaces\mathbf{Spaces}.

C wD C \stackrel{\simeq_w}{\to} D : a weak equivalence of ω\omega-categories, i.e. an ω\omega-functor essentially kk-surjective for all kk.

ωGrpd(Spaces)\omega\mathrm{Grpd}(\mathbf{Spaces}): ω\omega-categories with specified strict inverses for all kk-morphisms for all kk with inverse-assigning maps being morphisms in Spaces\mathbf{Spaces}.

ωGrp(Spaces)\omega\mathrm{Grp}(\mathbf{Spaces}) : ω\omega-groupoids of the form G=C(a,a)G = C(a,a) for CωGrpd(Spaces)C \in \omega\mathrm{Grpd}(\mathbf{Spaces}) and aObj(C)a \in Obj(C).

BG\mathbf{B}G: the unique ω\omega-groupoid with a single object \bullet and C(,)=GC(\bullet,\bullet) = G.

Then:

Definition [ω\omega-covers] For XωCat(Spaces)\mathbf{X} \in \omega\mathrm{Cat}(\mathbf{Spaces}), a cover of XX is an epimorphic weak equivalence π:Y> wX. \pi : \mathbf{Y} \stackrel{\simeq_w}{\to \gt} \mathbf{X} \,. A refinement of covers is an epimorphism of covers Y > Y π π X. \array{ \mathbf{Y} &\to \gt& \mathbf{Y}' \\ & {}_\pi\searrow \swarrow_{\pi'} \\ & \mathbf{X} } \,. Denote the category formed by covers of X\mathbf{X} by Covers(X)\mathrm{Covers}(\mathbf{X}).

Definition [ω\omega-cohomology] For X,CωCat(Spaces)\mathbf{X},C \in \omega\mathrm{Cat}(\mathbf{Spaces}), the cohomology of X\mathbf{X} with coefficients in CC is H(X,C):= YCovers(X)hom(Y,C). H(\mathbf{X},C) := \int^{\mathbf{Y} \in \mathrm{Covers}(\mathbf{X})} \mathrm{hom}(\mathbf{Y},C) \,.

cocycles - objects of H(X,C)H(\mathbf{X},C)

coboundaries - 1-morphisms in H(X,C)H(\mathbf{X},C); if C=B n1U(1)C = \mathbf{B}^{n-1}U(1) then these have the familiar form a+dλ=ba + d\lambda = b.

cohomology classes - equivalence classes in H(X,C)H(\mathbf{X},C)

Definition [various kinds of cohomologoes]

Nonabelian cohomology: GG an ω\omega-group and C:=BGC := \mathbf{B}G.

Equivariant cohomology: X\mathbf{X} not a discrete ω\omega-category

group cohomology: equivariant cohomology of a point, for instance for GG a group and AA an abelian group H n(G,A)=H(BG,B nA)H^n(G,A) = H(\mathbf{B}G, \mathbf{B}^{n}A).

differential cohomology : X=Π ω(X)\mathbf{X} = \Pi_\omega(X) (flat) or X=P n(X)\mathbf{X} = P_n(X) (fake-flat) – see next definition [there is also the non-fake flat case, but that requires more discussion]

Definition [fundamental ω\omega-groupoids]

For XSpacesX \in \mathbf{Spaces}, let Π ω(X)\Pi_\omega(X) be the ω\omega-groupoid whose kk-morphisms are maps of the standard kk-disk in XX, constant in a neighbourhood of the boundary of the disk and modulo thin-homotopy.

Denote the truncation of Π ω(X)\Pi_\omega(X) at nn-morphisms by P n(X)P_n(X) and the result of identifying in P n(X)P_n(X) nn-morphisms equivalence in Π ω(X)\Pi_\omega(X) by Π n(X)\Pi_n(X).

In particular P 0(X)P_0(X) is the discrete ω\omega-category over XX and Π 1(X)\Pi_1(X) the fundamental 1-groupoid of XX. (I had the notation for P 0(X)P_0(X) consistently wrong in the above comments. Sorry.)

Definition [geometric realization]

Let ||:ωCat(Spaces)Spaces |-| : \mathbf{\omega}\mathrm{Cat}(\mathbf{Spaces}) \to \mathbf{Spaces} be defined such that for CωCat(Spaces)C \in \omega\mathrm{Cat}(\mathbf{Spaces}) the realization |C||C| is the sheaf |C|:Uhom(Π ω(U),C). |C| : U \mapsto hom(\Pi_\omega(U),C) \,.

[“smooth homotopy hypothesis”]

The fundamental ω\omega-groupoid construction and realization fit together nicely (ahem) Π ω:SpacesωCat(Spaces):|| \Pi_\omega : \mathbf{Spaces} \leftrightarrow \omega\mathrm{Cat}(\mathbf{Spaces}) : |-|

Definition [differential forms on Spaces\mathbf{Spaces}]

Write Ω Spaces\Omega^\bullet \in \mathbf{Spaces} for the deRham sheaf Ω :UΩ (U). \Omega^\bullet : U \mapsto \Omega^\bullet(U) \,. For XSpacesX \in \mathbf{Spaces} write Ω (X):=hom(X,Ω ) \Omega^\bullet(X) := hom(X,\Omega^\bullet) for the space of differential forms on XX. This naturally carries the structure of a differential non-negatively graded-commutative algebra (“DGCA”) and yields a contravariant functor Ω :SpacesDGCAs. \Omega^\bullet : \mathbf{Spaces} \to DGCAs \,.

Definition [classifying space for algebra-valued differential forms]

For AA any DGCA, denote by S(A)S(A) the space given by the sheaf S(A):Uhom(A,Ω (X)). S(A) : U \mapsto hom(A,\Omega^\bullet(X)) \,. This yields a contravariant functor S:DGCAsSpaces. S : DGCAs \to \mathbf{Spaces} \,.

[“differential homotopy hypothesis”] The contravariant functors Ω :SpacesDGCAs:S \Omega^\bullet : \mathbf{Spaces} \leftrightarrow DGCAs : S form a contravariant adjunction with unit Id DGCAsΩ (S(A)) Id_{DGCAs} \Rightarrow \Omega^\bullet(S(A)) given by the inclusion AΩ (S(A)) A \hookrightarrow \Omega^\bullet(S(A)) that acts as (aA)U:((fhom(A,Ω (U)))(f(a)Ω (U))). (a \in A) \mapsto \forall U : ( (f \in hom(A,\Omega^\bullet(U))) \mapsto (f(a) \in \Omega^\bullet(U))) \,.

Definition [L L_\infty-algebroid]

A (finite rank) L L_\infty-algebroid is a manifold XX and a cochain complex gg concentrated in non-positive degree of finite rank (A:=C (X))(A := C^\infty(X))-modules together with a linear (over the ground field )degree +1 algebra derivation d: A g * A g * d : \wedge^\bullet_A g^* \to \wedge^\bullet_A g^* (where g *g^* is the dual complex (over AA), hence non-negatively graded) such that d 2=0. d^2 = 0 \,. The DGCA thus defined CE(g):=( A g *,d) CE(g) := (\wedge^\bullet_A g^*, d) is the Chevalley-Eilenberg DGCAof gg. We identify the category of L L_\infty-algebroids with the image CE:L DGCAs. CE : L_\infty \to DGCAs \,.

If gg is concentrated in the first nn degrees this is a Lie nn-algebroid. If X=ptX = pt this is a (finite dimensional) L L_\infty-algebra or a Lie nn-algebra if gg is concentrated in the first nn-degrees. If X=ptX = pt and d:g *g *g *d : g^* \to g^* \wedge g^* this is a dg-Lie algebra.

Definition [\infty-Lie integration and differentiation]

Combining the smooth and the differential “homotopy hypotheses” with this definition we obtain the situation ωCat(Spaces)SpacesDGCAsL . \omega\mathrm{Cat}(\mathbf{Spaces}) \leftrightarrow \mathbf{Spaces} \leftrightarrow DGCAs \leftarrow L_\infty \,. We say that going from right to left through this is \infty-Lie integration. Going from left to right is \infty-Lie differentiation.

Definition [L L_\infty-algebraic cocycle]

For gg a Lie nn-algebra, XX a space and Y>XY \to\gt X a regular epimorphism, a gg-cocycle on XX is a DGCA morphism

Ω vert (Y)CE(g):A vert. \Omega^\bullet_{vert}(Y) \leftarrow CE(g) : A_{vert} \,.

Let GG be the simply connected nn-group integrating gg, i.e. BG:=Π nSCE(g). \mathbf{B}G := \Pi_n \circ S \circ CE (g) \,. Notice that the integration of Ω vert (Y)\Omega^\bullet_{vert}(Y) is Π n vert(Y):=Π nS(Ω vert ), \Pi_n^{vert}(Y) := \Pi_n \circ S (\Omega^\bullet_{vert}) \,, the sub nn-groupoid of Π n(Y)\Pi_n(Y) of all those nn-paths that project down to a point in XX.

Definition [Lie integration of L L_\infty-algebraic cocycles]

Let Y\mathbf{Y} be a quotient of Π n vert(Y)\Pi_n^{vert}(Y) which is weakly equivalent to P 0(X)P_0(X). Let GG' be a quotient of GG such that there is a commuting diagram

Π n verty(Y) Π nSCE(A vert) BG Y g G w P 0(X). \array{ \Pi_n^{verty}(Y) &\stackrel{\Pi_n\circ S \circ CE (A_{vert})}{\to}& \mathbf{B}G \\ \downarrow && \downarrow \\ \mathbf{Y} &\stackrel{g}{\to}& \mathbf{G}' \\ \downarrow^{\simeq_w} \\ P_0(X) } \,. The existence of at least one such diagram is the integrability condition on A vertA_{vert}.

Then this nonabelian cocycle g:YBG g : \mathbf{Y} \to \mathbf{B}G' is a nonabelian cocycle integrating A vertA_{\mathrm{vert}}.

Proposition. This integration of L L_\infty-algebraic cocylces to nonabelian cocycles yields in particular a refinement (of the method and of the result) in nonabelian cohomology of the Chern-Simons cocycles constructed by Brylinski&MacLaughlin:

where B&M obtain Chern-Simons nn-cocycles obstructing lifts of GG-cocycles through shifted central extensions

B n1U(1)G^G \mathbf{B}^{n-1}U(1) \to \hat G \to G

this procedure constructs the corresponding twisted G^\hat G-cocycles which generalize the twisted bundles that appear in twisted K-theory.

Definition [shifted central extension of ω\omega-groups]

The ω\omega-functors B n1U(1)G^G \mathbf{B}^{n-1}U(1) \to \hat G \to G are a shifted central extension is the first one is monic, the second epic and if there is a weak equivalence from the weak quotient to GG: B(U(1)G^) wBG. \mathbf{B}(U(1) \to \hat G) \stackrel{\simeq_w}{\to} \mathbf{B}G \,.

Given a GG-cocycle YgBG \mathbf{Y} \stackrel{g}{\to} \mathbf{B}G we ask for the obstruction to lifting this through B nU(1) BG^ Y g BG. \array{ && \mathbf{B}^n U(1) \\ && \downarrow \\ && \mathbf{B}\hat G \\ && \downarrow \\ \mathbf{Y} &\stackrel{g}{\to}& \mathbf{B}G } \,. With Y\mathbf{Y} chosen fine enough we can always lift to a twisted G^\hat G-cocycle, namely a B(U(1)G^)\mathbf{B}(U(1) \to \hat G)-cocycle as in figure 4.

In [SSS] this is described for L L_\infty-algebraic cocycles. Hit everything in there with Π nS:DGCAsωCat(Spaces)\Pi_n \circ S : DGCAs \to \omega\mathrm{Cat}(\mathbf{Spaces}) to get the corresponding statements in nonabelian cohomology.

Posted by: Urs Schreiber on July 21, 2008 2:17 AM | Permalink | Reply to this

Re: Some ω-Questions

In the definition of “cover” what I really want to say, I suppose, is that π:YX \pi : \mathbf{Y} \stackrel{\simeq}{\to} \mathbf{X} a weak equivalence which is a fibration, i.e. an acyclic fibration.

I need to get a better understanding of the acyclic fibrations in ωCat\omega\mathrm {Cat} and those in ωCat(Spaces)\omega\mathrm{Cat}(\mathbf{Spaces}).

In the folk model structure on ωCat\omega\mathrm{Cat} I understand the cofibrations (p. 2) and the weak equivalences (p. 4) and I understand the standard fact (p. 4 here) how these two define the fibrations.

But I am lacking a good concrete understanding of what the fibrations in general and the acyclic fibrations in particular in ωCat\omega\mathrm{Cat} would be. I’d hope that my original idea – weak equivalences which are epic – would still survive as examples for acyclic fibrations or maybe even exhaust them. I suppose I should be able to check this. But not tonight.

Then I am a bit unsure about the same in ωCat(Spaces)\omega\mathrm{Cat}(\mathbf{Spaces}). Inside Spaces\mathbf{Spaces} there is Manifolds\mathbf{Manifolds} and if working in ωCat(Manifolds)\omega\mathrm{Cat}(\mathbf{Manifolds}) all surjective comoponent maps in a fibration should actually be surjective submersions. Is the contact with standard Morita morphisms of groupoids lost when I think of working in ωCat(Spaces)\omega\mathrm{Cat}(\mathbf{Spaces}) instead of ωCat(Manifolds)\omega\mathrm{Cat}(\mathbf{Manifolds})?

Well, lots of questions. I keep thinking about it…

Posted by: Urs Schreiber on July 23, 2008 10:03 PM | Permalink | Reply to this
Read the post Codescent and the van Kampen Theorem
Weblog: The n-Category Café
Excerpt: On codescent, infinity-co-stacks, fundamental infinity-groupoids, natural differential geometry and the van Kampen theorem
Tracked: October 21, 2008 9:27 PM

Post a New Comment