The n-Category Café

If $G$ is a topological group, what is a $K(G,1)$? Did you mean $BG$?

There are spaces where the fundamental group isn’t discrete (Hawaiian earring, say), but that’s getting a bit much :)

You’re saying there’s a god-given $\pi_n(G)$-valued $(n+1)$-cocycle on a sufficiently nice topological group G.

Instead of focusing on the specific cocycle, let’s think about its cohomology class. Then you’re saying that if G is sufficiently nice, there’s a god-given element of

…

where the square brackets mean ‘the set of homotopy classes of maps’.

Here $G$ is $(n-1)$-connected, so $K(G,1) = BG$ is $n$-connected with $\pi_{n+1}(BG) = \pi_n(G)$. So the first non-trivial stage in the Postnikov tower is

exactly the sort of map you are looking for.

I should add that I have the vague feeling you’re describing the Postnikov tower of $G$, or something like that.

I still haven’t quite found the time to think about this in the detail that I ought to, but it seems that this is a way of looking at the construction of higher String-like extensions by successively killing homtopy groups.

In fact, if I think about it for a bit it feels that I should be able to translate this directly to the construction of the strict String 2-group and of the strict Fivebrane 6-group by $L_\infty$-integration in those notes on Twisted nonabelian differential cohomology that some of you have seen.

One nice aspect of Christoph’s point of view, in contrast to the $L_\infty$-integration picture, is that it allows to talk not only about killing of non-torsion homotopy groups, but also about the torsion homotopy groups.

In particular, while I can express the steps $$Spin(n) \mapsto String(n) \mapsto Fivebrane(n)$$ in terms of $\infty$-Lie theory, I can’t look at it this way for the first two steps $$O(n) \mapsto S O(n) \mapsto Spin(n)$$ in the same way, since that requires killing two torsion groups ($\mathbb{Z}_2$). But I gather using Christoph’s point of view this is immediate:

As he points out, his 2-cocycle for $O(n)$ characterizes the extension $$\mathbb{Z}_2 \to Spin(n) \to S O(n) \,.$$ I suppose if we do a bit of “negative thinking” a la Toby Bartels we can tell a story how even one step further down his construction yields a 1-cocycle which characterizes $S O(n)$ with respect to $O(n)$. Hm…

As Hisham Sati taught me, there is indication that for the purpose of physics we need to go up to the 11-connected cover of $Spin(n)$ given by the strict 10-group to be called $Ninebrane(n)$. The NS-9-brane is the “space filling nine-brane at the Hořava-Witten-end-of-the-world” or the like, and one expects some curious properties, with its worldvolume theory being closely related to the target space theory of the heterotic string.

Hisham has some articles on the arXiv with tentative discussion of this point, and he tells me that some other group recently was able to make some aspects of these NS 9-branes more concrete.

In any case. To get to $Ninebrane(n)$ involves not just killing $\pi_{11}(O(n)) = \mathbb{Z}$ but first also the two torsion groups $\pi_{8}(O(n)) = \pi_{9}(O(n))= \mathbb{Z}_2$.

I’ll say some random stuff that might be slightly useful.

In group cohomology as traditionally formulated, a cochain is a map $f : G^n \to A$. Here $G$ is a group, and $A$ an abelian group on which $G$ acts. There’s no requirement that $f$ be continuous or smooth. Indeed, there’s no way to formulate such a requirement.

When $G$ and $A$ are topological groups and the action of $G$ on $A$ is continuous, we can require that our cochains $f$ be continuous. This gives a new kind of group cohomology with strikingly different properties. It’s usually called ‘van Est’ cohomology, since it was first studied here:

• W. T. van Est, On the algebraic cohomology concepts in Lie groups I & II Indag. Math. 17 (1955), 225–233, 286–294.

Similarly, when $G$ and $A$ are Lie groups (or more general smooth groups) and the action of $G$ on $A$ is smooth, there’s a kind of group cohomology where the cochains are required to be smooth.

This is often also called van Est cohomology, since van Est showed that for $G$ and $A$ Lie groups, the group cohomology defined with continuous cocycles matches the group cohomology defined with smooth cocycles. Both are very different from group cohomology as defined using arbitrary cocycles!

I became a bit of an expert on van Est cohomology when proving some no-go theorems for Lie 2-groups in HDA6, and you can find more information about it in Section 8.5 of that paper (the section called ‘2-groups from Chern–Simons theory’). Besides the original paper of van Est, there are also useful papers by Hu and Mostow:

• S. T. Hu, Cohomology theory in topological groups, Mich. Math. J. 1 (1952), 11–59.
• G. D. Mostow, Cohomology of topological groups and solvmanifolds, Ann. Math. 73 (1961), 20–48.

Later, Brylinski invented a subtler kind of group cohomology involving ‘locally smooth’ cochains. This seems to be better than van Est cohomology for the sort of stuff we’re interested in.

For example:

There’s a very interesting nontrivial 3-cocycle

$$a : G \to U(1)$$

when $G$ is a compact simply-connected simple Lie group acting trivially on $U(1)$. This gives rise to an interesting and by now rather famous 2-group with $a$ as associator.

On the other hand, every smooth or even continuous cocycle of the above sort is trivial. You can see a proof in Thm. 59 of HDA6.

On the third hand, there are very interesting nontrivial locally smoothable cocycles of the above sort.

This is related to how smooth anafunctors are more important than smooth functors. Smooth anafunctors are gadgets that can be locally but not globally described as smooth functors.

What do you mean by the ‘Moore–Postnikov tower of a pointed space’?

I just know about the Moore–Postnikov tower of a map

$$f : X \to Y$$

between pointed spaces. This is a sequence of spaces

$$X \to \cdots \to X_n \to X_{n-1} \to \cdots \to X_1 \to X_0 = Y$$

that interpolate between the space $X$ and the space $Y$. When $n = 0$, $X_n$ is just $Y$. Then, as $n$ increases, $X_n$ starts looking more and more like $X$ — ‘from the bottom up’.

A bit more precisely: the map $X_n \to X_{n-1}$ induces an isomorphism of homotopy groups except for the $n$th, at which point I guess it looks exactly like the map

$$f_* : \pi_n(X) \to \pi_n(Y)$$

Now, the Postnikov tower of a pointed space $X$ is just the Moore-Postnikov tower of the map

$$X \to *$$

We start out with the point $*$ and get a tower of spaces with more and more homotopy groups matching those of $X$.

I’m hoping that what you’re calling the ‘Moore–Postnikov tower of a pointed space’ is what I’d call the Moore–Postnikov tower of the map

$$* \to X$$

Here we start out with $X$ and get a tower of spaces with more and more homotopy groups matching those of a point. That seems to fit what you’re saying.

Alas, I haven’t figured out how this Moore–Postnikov stuff fits with what I was calling cokilling cohomology groups. I think they agree in the special case of going from $Spin(n)$ to $String(n)$, but I’m not sure how generally they agree.

In other words: can we always build the successive stages of what I’m calling the Moore–Postnikov tower of the map

$$* \to X$$

by successively cokilling the cohomology groups of $X$, from the bottom up?

for n > 2, calling this an group is true
but in a superficial and misleading sense.

I think it’s much more relevant that this gives 2-stages, i.e. a fibration and not an iterated fibration

Todd,

thanks for your message. I know that I am a bad student of yours. I should already know this stuff much better from our previous conversations about related things. I hope you can bear with me.

You had taught me about Day convolution here:

For $C$ a monoidal $V$-enriched category, presheaves $F,G : C^{op} \to V$ inherit the Day convolution tensor product defined as $$(F \star G)(-) = \int^{c,d \in C} F(c)\otimes G(c) \otimes hom(-, c \otimes d) \,.$$

This generalizes the familiar convolution product of functions which we recover by taking $C$ to be a discrete monoidal category, i.e. a monoidal set, i.e. a monoid, and regard for instance $F,G : C^{op} \to Set$ as categorified $\mathbb{N}$-valued functions on this monoid, in which case their Day convolution product is the ordinary convolution product of (categorified) $\mathbb{N}$-valued functions $$(F \star G)(e) = \int^{c,d} F(c) \times G(d) \times hom_C(e,c \cdot d) = \oplus_{c\cdot d = e} F(c)\times G(d) \,.$$

The presheaf $I$ constant on the tensor unit $I$ in $V$ is indeed the tensor unit under the Day convolution product since — let me see $$\begin{aligned} (F \star I)(e) &= \int^{c,d \in C} F(c)\otimes hom(e, c \otimes d) \\ &= \int^{c\in C} F(c)\otimes hom(e, c ) \\ &= F(e) \end{aligned} \,,$$ I suppose, where about the step from the first to the second line I feel a bit shaky, while the step from the second to the third line is Yondeda reduction as you also explained here.

Okay, now with $C = \Box$ the cubical category and $V = Set$, we have underlying any $\omega$-categories $C,D$ their cubical nerves $$N C = Hom( O([-]), C) : \Box^{op} \to Set$$ which inherit the Day convolution product from the standard monoidal product on $\Box$ and the Crans-Gray tensor product is supposed to be a coequalizer from the free $\omega$-category on that, which is $$\begin{aligned} F(N C \star N D) &= \int^n (N C \star N D)([n])\cdot O([n]) \\ &= \int^{n,p,q} Hom(O([p]),C)\times Hom(O([q]),D) \times Hom([n],\underbrace{[p]\otimes [q]}_{[p+q]})\cdot O([n]) \\ &\simeq \int^{p,q} Hom(O([p]),C)\times Hom(O([q]),D) \cdot O([p+q]) \end{aligned} \,,$$ where in the last step I did the coend over $n$ by Yoneda.

Now I need to figure out which two morphisms into this $F(N C \star N D)$ are coequalized by $C \otimes_{CransGray} D$.

Hm, so I am supposed to use that $N \circ F$ is monoidal, so that for instance $$F \circ N \circ F(N C \star N D) \simeq F (N \circ F \circ N (C) \star N \circ F \circ N (C) ) \,.$$ Hm, so I should use the counit in two places…

Perhaps a word of caution is needed here to any passers-by of this café. It seems that here Todd is using MacLane’s categorical convention for the objects of $\Delta$, so for him $[p] = \{1,2, ... , p\}$ with the usual order.

On the other hand within sources on simplicial homotopy theory, it is more usual to write $[p] = \{0,1,2, ... , p\}$ (obvious order) and then $[p]\oplus[q]$ looks like $[p + q + 1]$, but beware it is a functor of two variables which sometimes further confuses people. Thus for simplicial homotopy $[p]$ does have the right dimension.

In the past, I have wasted quite a lot of time when reading sources on this sort of thing because I did not verify which indexing was being used. Once that is sorted the calculations are fairly routine, but clearly can get confusing if you don’t check up first!

I wrote #:

Now I need to figure out […]

I should do the toy case of abelian groups first, as Todd suggested.

So let $C = \mathbf{B}\mathbb{Z}$ be the category with a single object and $\mathbb{Z}$ worth or morphisms. This is monoidal (it’s a 2-group even) with the standard group operation on the morphisms.

An abelian group $A$ is in particular a $\mathbb{Z}$-module hence a presheaf on $C$ $$U(A) : C^{op} \to Sets \,.$$

So for $A$, $B$ two abelian groups with underlying presheaves $U A$ and $U B$ with underlying sets which I’ll also write $U A := U A(\bullet)$ and $U B := U B(\bullet)$, their Day convolution product $$(U A \star U B)(-) = \int^{\bullet,\bullet \in \mathbf{B}\mathbb{Z}} U A(\bullet) \times U A(\bullet) \times hom_{\mathbf{B}\mathbb{Z}}(-,\bullet)$$ is the universal co-extraordinary family in that for any morphism $\bullet \stackrel{k}{\to} \bullet$ in $C = \mathbf{B}\mathbb{Z}$ we have the cone $$\array{ U A \times U B \times \mathbb{Z} &\stackrel{(k\cdot-)\times Id \times (-+k)}{\to}& U A \times U B \times \mathbb{Z} \\ \downarrow^{Id \times (k \cdot -)\times (-+k)} && \downarrow \\ U A \times U B \times \mathbb{Z} &\to& \int^{\bullet,\bullet} U A(\bullet) \times U B(\bullet) \times hom_{\mathbf{B}\mathbb{Z}}(\bullet,\bullet) } \,.$$ That should mean that $(U A \star U B)(\bullet) = (U A(\bullet) \otimes_\mathbb{Z} U B(\bullet)) \times \mathbb{Z}$.

Ah, so the coequalizer I am looking for needs to kill this factor of $\mathbb{Z}$.

Okay, I ended my last message # essentially with guessing that I am to coequalize the maps

$$F \circ U \circ F (U A \star U B) \stackrel{ \epsilon \circ Id_{F(U A \star U B)} }{\to} F (U A \star U B)$$ and $$F \circ U \circ F (U A \star U B) \stackrel{\simeq}{\to} F( U \circ F \circ U A \star U \circ F \circ U A ) \stackrel{ F ( \epsilon \circ Id_{U A} \star \epsilon \circ Id_{U B} ) }{\to} F (U A \star U B) \,,$$ where $\epsilon$ is supposed to denote the counit of the monoidal adjunction $$\epsilon : F \circ U \to Id \,.$$

This is becoming notationally a bit of a challenge now (for me at least), hope the above is readable.

Is the coequalizer of these two maps really $A \otimes_{\mathbb{Z}} B$? Looks not unplausible. The first map just plays around with free constructions, while the second actually knows the group structure on $A$ and $B$ but leaves the superfluous $\mathbb{Z}$ alone. So that might just drop out…