## July 18, 2008

### Some ω-Questions

#### Posted by Urs Schreiber

I have some questions on $\omega$-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 $\omega\mathrm{Cat}$ an $\omega$-functor $F : C \stackrel{\simeq_w}{\to} D$ between $\omega$-categories is a weak equivalence if it is essentially $k$-surjective for all $k \in \mathbb{N}$ in the sense of definition 4 of Baez-Shulman.

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

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

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

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

Consider the standard Gray-like closed monoidal structure on $\omega\mathrm{Cat}$ and consider the category $\omega\mathrm{CoSimp} := Funct(\Delta, \omega\mathrm{Cat})$ of cosimplicial $\omega$-categories. It seems to me that using the inner hom in $\omega\mathrm{Cat}$ degreewise cosimplicial $\omega$-categories become naturally enriched over $\omega\mathrm{Cat}$. 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 $\omega\mathrm{Cat}$-enrichment of $\omega\mathrm{CoSimp}$? Has this been stated anywhere in the literature? What’s the best formal way to characterize it?

Question 3) descent $\omega$-categories

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

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

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 $\omega\mathrm{Cat}$ to move the $\omega$-glob around.

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

Does that sound right?

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

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

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

for all $C$.

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

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

$\Pi_0^Y(X) \stackrel{\simeq_w}{\to} \Pi_0(X) \,.$

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 $C$-valued nonabelian cohomology

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

the colimit on the right with something going out of all $\omega$-categories weakly equivalent to $\Pi_0(X)$ 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

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## 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 $\infty\mathbf{Groupoid}$ and pick a notion of $\mathbf{Spaces}$ 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 $\infty\mathbf{Groupoids} \simeq \mathbf{Spaces}$.

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

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

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

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

Second subquestion: we have Quillen model category structures on both sides. Do $|-|$ and $\Pi_\omega$ 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: João Faria Martins on July 18, 2008 4:58 PM | Permalink | Reply to this

### Re: Some ω-Questions

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

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

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 $\omega$-groupoid $\Pi_\omega(X)$ in my sense.

There are two issues here:

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

$P X : U \mapsto hom(U\times I, X)$

then the space of paths modulo thin homotopy is the sheaf

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

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

b) How does composition in $\Pi_\omega(X)$ 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 $P_1(X)$ as defined for instance here.

We get a double-category from that by considering $P_1(Mor(P_1(X)))$. 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: João 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 $P$ of a principal 1-bundle $P \to X$. This would translate to the usual cocycles by taking $Y := P$ as the surjective submersion. But then for nonabelian gerbes/2-bundles the connection data is

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

(satisfying potentially fake-flatness but nothing else)

and then

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

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

$(f) \in \Omega^0(Y \times_X Y \times_X Y, E)$.

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 $n$-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.

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 $X$ to be certain 1-paths in path space $P X$ 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: João 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 $p$-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 $C^*$ 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: João 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 $k$, then clearly all $k$-forms pulled back along it vanish. Conversely, if all these pulled back $k$-forms vanish then then rank has to be less than $k$, since if it were not, we could construct a $k$-form which picks up the contribution of the $k$-linearly independent tangent vecors in a neighbourhood of a point where the rank of the map is $k$.

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 $\omega$-functor $!:\mathcal{O}_n\rightarrow \text{pt}$, from the $n^{\text{th}}$ oriental to the point, is a weak equivalence in the folkloric model category structure on $\omega-\text{Cat}$.

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

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

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

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

Of course, the point $\text{pt}$ as an $\omega$-category consists only of a single $0$-cell $\bullet$ along with its identity cell at each dimension. So it follows that $!$ maps both of $s_{n-1}(\alpha_n)$ and $t_{n-1}(\alpha_n)$ to the (unique) identity on $\bullet$ at dimension $(n-1)$. Furthermore, it also follows that the identity on $\bullet$ at dimension $n$ provides us with a n $n$-cell whose $(n-1)$-source and $(n-1)$-target are both equal to $!(s_{n-1}(\alpha_n))=!(t_{n-1}(\alpha_n))$.

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

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 $\omega$-category which is weakly equivalent to an $\omega$-groupoid is itself an $\omega$-groupoid. However, in a precise sense $\mathcal{O}_n$ is about as far from being an $\omega$-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 $\mathcal{O}_n$.

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 $O_n$s are mapped into $\omega$-groupoids.

Please allow me to try to fix my question:

I assume just as $O_n$ is the free $\omega$-category on the $n$-simplex, we can also consider $U_n$ (does this have an established symbol?), the free $\omega$-groupoid on the $n$-simplex (having strict inverses for all $k$-morphisms for all $k$).

Is it right that these $U_n$ 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 $\omega$-groupoid $\Pi_0^Y(X)$ which I mention in the above entry is weakly equivalent to the discrete $\omega$-category over the space $X$, which I denoted $\Pi_0(X)$.

But the $k$-cells in $\Pi_0^Y(X)$ are built from “orientals in $Y^\bullet$” and I think that unless we restrict in the universal property defining it the $\omega$-categories $C$ to be $\omega$-groupoids, then $\Pi_0^Y(X)$ will fail to be weakly equivalent to $\Pi_0(X)$ 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 $C$ a strict monoidal category and $\mathbf{B}C$ the corresponding strict one-object 2-category we have for the degenerate space and its degenerate cover $(Y \to X) := (pt \to pt)$ that $Desc( Y^\bullet, hom(\Pi_0(-),\mathbf{B}C) ) = Monoids(C) \,.$

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

This should amount to adding to the definition of $O(\Delta^3)$, 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 $O(\Delta^n)$ 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 $\omega$-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 $W$ between small categories such that

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

(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 $\omega$-version of this, and see if there is a fundamental localiser containing the funtors $O_n \to pt$ ($O_n$ the $n^{th}$ oriental), or at least characterise it.

The other thing would be to consider $\mathbf{O}$-sets (or $\mathbf{O}$-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 $\omega$-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 $S$, of an $\omega$-category $P_\omega(S)$ that behaves like the fundamental $\omega$-category of $S$ regarded as a discrete contractible space.

By construction, such $P_\omega(S)$s are weakly equivalent to the point. Moreover, I think by restricting to $S = [n] = \{0,1,2, \cdots, n\}$ they do arrange themselves in a cosimplicial $\omega$-category $P_\omega : \Delta \to \omega\mathrm{Cat}$ and hence induce a notion of cohomology. I am not sure for $n \gt 2$, but up to $n=2$ 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 $P_\omega$ 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 $0 \to 1 \to 2$. This forces product and coproduct to be invertible morphisms, after all.

So proper Frobenius structures in 2d appear this way only from $n=3$ 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 $\omega$-Cat. This is a non-symmetric tensor which has left and right closures, which we will denote by $\text{lax}_l(\mathbb{C},\mathbb{D})$ and $\text{lax}_r(\mathbb{C},\mathbb{D})$ respectively. We adopt this notation because these are precisely the appropriate (left and right handed) generalisations of the 2-categories of $2$-functors, lax-natural transformations and modifications of the 2-dimensional theory.

Now, although $\otimes$ is not symmetric (or braided) as a monoidal structure on $\omega$-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 $\omega$-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 $\text{lax}_r$ to enrich $\omega$-Cat over itself. Notice here that were we to use $\text{lax}_l$ for this we would not obtain a category enriched wrt $\otimes$, but would instead obtain a category enriched wrt the reverse tensor $X\otimes^\circ Y = Y\otimes X$.

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

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

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

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

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

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

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

BTW On the topic of the Gray tensor product of strict $\omega$-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 $\omega$-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 $\omega$-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 $\omega$-functors $\mathcal{O}(G^n\otimes\Delta^m)\cong \mathbb{G}^n\otimes\mathcal{O}(\Delta^m) \rightarrow\mathcal{E}^m$ which are natural in $m\in\Delta$.”

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:

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

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

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

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

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

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

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

Then:

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

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

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

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

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

Definition [various kinds of cohomologoes]

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

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

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

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

Definition [fundamental $\omega$-groupoids]

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

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

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

Definition [geometric realization]

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

[“smooth homotopy hypothesis”]

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

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

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

Definition [classifying space for algebra-valued differential forms]

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

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

Definition [$L_\infty$-algebroid]

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

If $g$ is concentrated in the first $n$ degrees this is a Lie $n$-algebroid. If $X = pt$ this is a (finite dimensional) $L_\infty$-algebra or a Lie $n$-algebra if $g$ is concentrated in the first $n$-degrees. If $X = pt$ and $d : g^* \to g^* \wedge g^*$ 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 $\omega\mathrm{Cat}(\mathbf{Spaces}) \leftrightarrow \mathbf{Spaces} \leftrightarrow DGCAs \leftarrow L_\infty \,.$ We say that going from right to left through this is $\infty$-Lie integration. Going from left to right is $\infty$-Lie differentiation.

Definition [$L_\infty$-algebraic cocycle]

For $g$ a Lie $n$-algebra, $X$ a space and $Y \to\gt X$ a regular epimorphism, a $g$-cocycle on $X$ is a DGCA morphism

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

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

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

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

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

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

Proposition. This integration of $L_\infty$-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 $n$-cocycles obstructing lifts of $G$-cocycles through shifted central extensions

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

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

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

The $\omega$-functors $\mathbf{B}^{n-1}U(1) \to \hat G \to G$ 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 $G$: $\mathbf{B}(U(1) \to \hat G) \stackrel{\simeq_w}{\to} \mathbf{B}G \,.$

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

In [SSS] this is described for $L_\infty$-algebraic cocycles. Hit everything in there with $\Pi_n \circ S : DGCAs \to \omega\mathrm{Cat}(\mathbf{Spaces})$ 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 $\pi : \mathbf{Y} \stackrel{\simeq}{\to} \mathbf{X}$ a weak equivalence which is a fibration, i.e. an acyclic fibration.

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

In the folk model structure on $\omega\mathrm{Cat}$ 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 $\omega\mathrm{Cat}$ 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 $\omega\mathrm{Cat}(\mathbf{Spaces})$. Inside $\mathbf{Spaces}$ there is $\mathbf{Manifolds}$ and if working in $\omega\mathrm{Cat}(\mathbf{Manifolds})$ 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 $\omega\mathrm{Cat}(\mathbf{Spaces})$ instead of $\omega\mathrm{Cat}(\mathbf{Manifolds})$?

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

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