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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 8th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIn a comment to a MO [question](http://mathoverflow.net/questions/96191/operator-theoretical-models-for-k-mathbbz-3) on $K(\mathbb{Z}, 3)$ we get referred to another question, where André Henriques [points out](http://mathoverflow.net/questions/44045/naturally-occuring-k-pi-n-spaces-for-n-geq-2/46634#46634) that $Out(M)$ is (conjecturally) a model, where $M$ is a hyperfinite type III factor. In view of the ideas at [[outer automorphism]], isn't it more natural to think of the automorphism 2-group of $M$? I wonder if that would satisfy John Baez's quest from [TWF149](http://math.ucr.edu/home/baez/week149.html): >How about K(Z,3)? Well, I don't know a nice geometrical description of this one. And this really pisses me off!...the integers, the group U(1), and infinite-dimensional complex projective space are all really important in quantum theory. This is perhaps more obvious for the latter two spaces - the integers are so basic that it's hard to see what's so "quantum-mechanical" about them. However, since each of these spaces is just the loop space of the next, they're all part of tightly linked sequence... and I want to know what comes next!

   In a comment to a MO question on $K(\mathbb{Z}, 3)$ we get referred to another question, where André Henriques points out that $Out(M)$ is (conjecturally) a model, where $M$ is a hyperfinite type III factor. In view of the ideas at outer automorphism, isn’t it more natural to think of the automorphism 2-group of $M$?

   I wonder if that would satisfy John Baez’s quest from TWF149:

   How about K(Z,3)? Well, I don’t know a nice geometrical description of this one. And this really pisses me off!…the integers, the group U(1), and infinite-dimensional complex projective space are all really important in quantum theory. This is perhaps more obvious for the latter two spaces - the integers are so basic that it’s hard to see what’s so “quantum-mechanical” about them. However, since each of these spaces is just the loop space of the next, they’re all part of tightly linked sequence… and I want to know what comes next!

2. • CommentRowNumber2.
   • CommentAuthorAndrew Stacey
   • CommentTimeMay 8th 2012
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownItexDoes my answer to the question linked in the question linked suffice? $K(\mathbb{Z},2)$ is the projective unitary group and that's a group so we take its classifying space. There is a specific model for this: let $H$ be a Hilbert space and $\mathbb{P} U(H)$ its projective unitary group. Let $\mathcal{H}$ be the space of Hilbert-Schmidt operators on $H$. This is again a Hilbert space. There is an action of $U(H)$ on $\mathcal{H}$ by conjugation, and the centre of $U(H)$ acts trivially so this is really an action of $\mathbb{P}U(H)$. This defines an action of $\mathbb{P}U(H)$ on the contractible space $U(\mathcal{H})$ and the quotient is then $K(\mathbb{Z},3)$. Since $U(H)$ acts on $\mathcal{H}$ as the "inner" automorphisms, the quotient is the "outer" automorphisms. I guess this is what André was referring to, though I don't get what's conjectural about the construction.

   Does my answer to the question linked in the question linked suffice? $K(\mathbb{Z},2)$ is the projective unitary group and that’s a group so we take its classifying space. There is a specific model for this: let $H$ be a Hilbert space and $\mathbb{P} U(H)$ its projective unitary group. Let $\mathcal{H}$ be the space of Hilbert-Schmidt operators on $H$. This is again a Hilbert space. There is an action of $U(H)$ on $\mathcal{H}$ by conjugation, and the centre of $U(H)$ acts trivially so this is really an action of $\mathbb{P}U(H)$. This defines an action of $\mathbb{P}U(H)$ on the contractible space $U(\mathcal{H})$ and the quotient is then $K(\mathbb{Z},3)$. Since $U(H)$ acts on $\mathcal{H}$ as the “inner” automorphisms, the quotient is the “outer” automorphisms. I guess this is what André was referring to, though I don’t get what’s conjectural about the construction.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 8th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn André's construction there is a group whose underlying homotopy type is a $K(\mathbb{Z},3)$.

   In André’s construction there is a group whose underlying homotopy type is a $K(\mathbb{Z},3)$.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 9th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAndré says that the conjectural part is whether the automorphism group of the hyperfinite type III factor is contractible. What do you gain by having your Eilenberg-MacLane space be a group?

   André says that the conjectural part is whether the automorphism group of the hyperfinite type III factor is contractible.

   What do you gain by having your Eilenberg-MacLane space be a group?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 9th 2012
   • (edited May 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> What do you gain by having your Eilenberg-MacLane space be a group? This is the implicit open question that people interested in this discussion here need to answer. It really only becomes an interesting question if you add some desiderata. Because, if you are just interested in a simple construction that gives you the homotopy type $K(\mathbb{Z},n)$ realized as a topological group, that's trivial and ancient: do $|DoldKan(\mathbb{Z}[n])|$. So the question here must be something else. As in the title of that MO-thread: can you give an "_operator theoretic_ topological group" whose underlying homotopy type is that of a $K(\mathbb{Z},n)$? Since people are referring to John Baez's old remarks on this: he observes that for the second and third $n$ the canonical such groups play a role in quantum physics. Okay, so now the question is: do we take two data points to be enough to quiver with excitement about a bold conjecture: that also for higher $n$ there are interesting quantum field theoretic groups at work. Or do we just shrug and say that in low dimension many things become related "by lack of space to be otherwise" that are not really conceptually related. I don't know. But if you are interested in that old "quest" of John's cited above one should maybe make explicit what the rules of the game being played are supposed to be.

   What do you gain by having your Eilenberg-MacLane space be a group?

   This is the implicit open question that people interested in this discussion here need to answer. It really only becomes an interesting question if you add some desiderata.

   Because, if you are just interested in a simple construction that gives you the homotopy type $K(\mathbb{Z},n)$ realized as a topological group, that’s trivial and ancient: do $|DoldKan(\mathbb{Z}[n])|$.

   So the question here must be something else. As in the title of that MO-thread: can you give an “operator theoretic topological group” whose underlying homotopy type is that of a $K(\mathbb{Z},n)$?

   Since people are referring to John Baez’s old remarks on this: he observes that for the second and third $n$ the canonical such groups play a role in quantum physics. Okay, so now the question is: do we take two data points to be enough to quiver with excitement about a bold conjecture: that also for higher $n$ there are interesting quantum field theoretic groups at work. Or do we just shrug and say that in low dimension many things become related “by lack of space to be otherwise” that are not really conceptually related.

   I don’t know. But if you are interested in that old “quest” of John’s cited above one should maybe make explicit what the rules of the game being played are supposed to be.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 9th 2012
   • (edited May 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI should maybe add for clarification: I find André's suggestion most noteworthy. The hyperfinite type $III_1$ factor is the hallmark of 2-dimensional QFT and Connes has argued since long that it's its outer automorphisms that are important for the quantum physics. So this suggestion is a great continuation of John's "quest", I think. If the conjecture is right.

   I should maybe add for clarification:

   I find André’s suggestion most noteworthy. The hyperfinite type $III_1$ factor is the hallmark of 2-dimensional QFT and Connes has argued since long that it’s its outer automorphisms that are important for the quantum physics. So this suggestion is a great continuation of John’s “quest”, I think. If the conjecture is right.

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 9th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIs there anything to the thought that whenever you see outer automorphisms mentioned, really you should think of the 2-group of which it is the truncation?

   Is there anything to the thought that whenever you see outer automorphisms mentioned, really you should think of the 2-group of which it is the truncation?

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMay 9th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexMight be useful depending on what is actually going on. For me, this is one way to define the notion of outer outomorphisms for higher groups. There may be other useful definitions. Defining for an $n$-truncated $A_\infty$-monoid $A$ $Out(A)$ to be the $n$-truncation of the (n+1)-group $Aut(\mathbf{B}A)$ is the right thing to do if one is interested in structures that locally look like the delooping $\mathbf{B}A$ ($A$-$n$-gerbes). Here in the present context, I wish I had more datapoints to see what it is we are trying to achieve.

   Might be useful depending on what is actually going on. For me, this is one way to define the notion of outer outomorphisms for higher groups. There may be other useful definitions. Defining for an $n$-truncated $A_\infty$-monoid $A$ $Out(A)$ to be the $n$-truncation of the (n+1)-group $Aut(\mathbf{B}A)$ is the right thing to do if one is interested in structures that locally look like the delooping $\mathbf{B}A$ ($A$-$n$-gerbes).

   Here in the present context, I wish I had more datapoints to see what it is we are trying to achieve.