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Let $G$ be a group, and let $L G = B G^{S^1}$ be the functor-groupoid into $B G$ out of $S^1 = B\mathbb{Z}$. So the objects of $L G$ are elements of $G$ and its morphisms are conjugations. An $n$-simplex in the nerve of $L G$ is a string of $n$ conjugations, which means it is determined by $n+1$ elements of $G$. Therefore, the free abelian group on the $n$-simplices in the nerve of $L G$ is isomorphic to $\mathbb{Z}[G^{n+1}] \cong \mathbb{Z}[G]^{\otimes (n+1)}$. This strongly suggests that the chain complex obtained from the nerve of $L G$ should be the Hochschild complex of $\mathbb{Z}[G]$, but making this identification extend to all the face maps seems to require a clever choice of the isomorphism between the $n$-simplices in $N(L G)$ and $G^{n+1}$, and I haven’t yet managed to be sufficiently clever. Does this exist anywhere? Does anyone know the answer?
This reminds me of ideas from cyclic homology rather than Hochschild. I used to know this:-( Look in Loday’s book mentioned in the Wikipedia entry on cyclic homology. Also Inventiones Mathematicae Volume 87, Number 2, 403-423, DOI: 10.1007/BF01389424, the paper is already listed in the references for this page.)
This is by John Jones and discusses cyclic homology and equivariant S^1 homology. (I remember some other papers but have forgotten their authors :-( I can ferret a bit more if you want. :-)
As I learned from Bressler in 2002, the cyclic nerve of a groupoid is canonically isomorphic as a cyclic set to the simplicial nerve of the inertia groupoid of the original groupoid (with its canonical cyclic structure). Inertia groupoid is a model for loop groupoid (in discrete situation).
@Zoran I note you use the term inertia groupoid, can you (or anyone else) provide a stub for this and also for the related inertia stack. This is mentioned in orbifold but no explanation is given. I am interested as it links in with the analysis of HQFTs and R. Kaufmann’s ideas (Which I should add to the page on crossed $G$-algebras!)
Why do you say it looks like cyclic homology? I understand cyclic homology even less than I understand Hochschild homology, but this chain complex looks to me more like Hochschild homology; there is no quotienting by the circle action. The nLab page cyclic homology says that Hochschild homology is the (co)homology of free loop space objects, which this definitely is. And also, I arrived at this object by a general construction (categorified trace in a bicategory) which I know produces Hochschild homology in other cases.
@Mike,
I only said that it reminded me of … . I do not understand cyclic homology either. :-(
You said:
but making this identification extend to all the face maps seems to require a clever choice of the isomorphism between the $n$-simplices in $N(L G)$ and $G^{n+1}$
can you be more specific about the problem? Without hearing what you tried I would probably make exactly the same choices and fail just as easily! One thought is that the construction looks a bit like that of the factorisation category so there may be something in the work of Baiues and Wirshing. They do discuss something about Hockschild in one of the papers they wrote on this.
I wasn’t specific, because I was hoping that someone would just say “oh, that’s well known” and give the answer or a reference. Zoran’s comment 3 seems like it might do the trick, if “inertia groupoid” and “cyclic nerve” mean what they might; Zoran, can you give a reference or describe the isomorphism?
Anyway, here’s what I was trying to do. The 0-simplices of N(LG) are elements of LG, i.e. loops in BG, i.e. elements of G, so no problem there. Its 1-simplices from g to h are pairs (k,m) such that g=km and h=mk. It seems obvious to identify this with the pair (k,m), and then the two face maps in N(LG) (being source and target) do exactly the right thing to be the Hochschild complex (multiply in the two orders).
It’s the 2-simplices that start to cause problems. A 2-simplex in N(LG) is a quadruple (k,m,k’,m’) such that mk = k’m’ – here (k,m) goes from km to mk, then (k’,m’) goes from mk=k’m’ to m’k’, and the composite (kk’,mm’) goes from km to m’k’. The three face maps in N(LG) send (k,m,k’,m’) to (k’,m’), (kk’,m’m), and (k,m).
The condition mk = k’m’ means that each of k,m,k’,m’ is determined uniquely by the others, so the set of 2-simplices in N(LG) is isomorphic to $G^3$, but what should the isomorphism be? The face maps acting on a triple (x,y,z) in the Hochschild complex take it to (y,zx), (xy,z), and (x,yz) respectively.
One obvious thing to try is to say that (k,m,k’,m’) corresponds to the triple (x,y,z)=(k,k’,m’m). Then the middle face map does the right thing ((xy,z) becomes (kk’,m’m) as it should), but the other two are wrong: the first Hochschild face map would send (k,k’,m’m) to (k’,m’mk) = (k’,m’k’m’), which is kind of like (k’,m’) but different. Same for the third face map.
I then thought of something like multiplying each element in the Hochschild triple on the right or left by one of the 0-simplices involved, but the only way I could get the first and third face maps to work out in this way resulting in messing up the second face map.
That’s about as far as I’ve gotten.
I mentioned Baues-Wirsching cohomology, and there are ideas there that look very similar. Also there is a lovely old preprint by charlie Wells, (there is a link from that entry (loc cit), but also here. I have written up a modern version of this (initially independently of Wells then I remembered its existence and found my tattered copy!). The problem came up in looking at ’track categories’ i.e. groupoid enriched categories, and so may be relevant to your question. (I can send you my notes on this if you think it would help.)
Mike, I am mainly busy for the next few hours so I can answer later; I have sent you one secret document which may help before that, however.
For relations between inertia groupoid and loop groupoid, look also into
Some old works of Goodwillie may also be helpful, e.g.
The theorem of Hinich (and the independent version of myself) on Drinfeld double is partly motivated by this picture.
Stub for inertia orbifold with redirects inertia groupoid, inertia stack. For certain quotients (by an action of a group) appearing in number theory the inertia terminology was used in the first half of the 20th century. It is probably related terminology.
A good quick survey of issues related to loop spaces, including some relations in section 3 dedicated to the Hochschild/cyclic homology, is in
Thanks, Zoran. That is very nice and clear. It is strange that I had not met it. Why ‘inertia’? It seems a very odd name for this.
As Zoran says, the term ’inertia group’ occurs in algebraic number theory. I first met the term while reading Lang’s Algebraic Number Theory, in the following context: let $E/k$ be a finite Galois extension of number fields; let $\mathfrak{p}$ be a prime ideal in the integers of $k$, and let $\mathfrak{B}$ be a prime ideal over $\mathfrak{p}$ in the integers of $E$. The Galois group permutes the primes over $\mathfrak{p}$, but if a Galois group element $g$ doesn’t move the prime $\mathfrak{B}$ at all – if the prime just sits still under the action of $g$ – we say $g$ belongs to the inertia group at $\mathfrak{B}$. Just old-fashioned terminology for the stabilizer subgroup.
Jim: I meant the groupoid $B G$, and I think I only referred to the Hochschild complex of $\mathbb{Z}[G]$ which is an algebra. Where did I talk about composing free loops?
but that does not really give why inertia groupoid is used here.
By the way, Zoran’s secret document (why is it secret?) looks like it answers my original question, but I haven’t digested it yet. As soon as I do, I’ll explain the answer here, and probably record it in a noncommutative section of Hochschild homology.
@Tim #15: were you referring to comment 13? I think of it this way: elements of inertia groups of a group action $G \times P \to P$ as are the same thing as loops in the action groupoid. These give you the objects of the inertia groupoid $(p, g: p \to p)$ corresponding to that action. To get from one inertia group $G_p$ to another $G_q$, you conjugate by an element $u: p \to q$. Just as in the definition of morphism in the inertial groupoid.
Mike asked
why is it secret?
The author of that document felt that there was nothing new there, at least as far as real experts are considered (like Beilinson…) so he never went into publishing it, despite much time and fascination he spent about the subject and the fact that other people later than that publication published smaller observations in the same subject. I tried to persuade him to finish and publish it but without success.
I have created (a bibliography) entry topological cyclic homology which also redirects topological Hochschild homology.
Todd, thanks for detailing the likely historical terminology connection further.
Here’s the answer; as I suspected, I just wasn’t being clever enough. Think of a morphism $g\to h$ in $L G$ as a single $k$ such that $h = k^{-1} g k$. Then an $n$-simplex in $N(L G)$ is determined by the first domain, $g_0$, and a sequence $(k_1,k_2,\dots,k_n)$, where we can recover the other objects as $g_1 = k_1^{-1} g_0 k_1$, etc. We send this to the cyclic sequence $(k_1,k_2,\dots,k_n, k_n^{-1} \cdots k_1^{-1} g_0)$. Then the middle face maps which just compose two of the $k$s are obviously correct, since $(k_i k_{i+1})^{-1} = k_{i+1}^{-1} k_i^{-1}$, the rightmost face map cancels $k_n$ with $k_n^{-1}$ (corresponding to forgetting $k_n$ in the original $n$-simplex), and the leftmost face map condenses $k_1^{-1} g_0 k_1$ to $g_1$, the new first domain after we forget $k_1$ in the original $n$-simplex.
That looks right and I feel I have seen it before.
@Todd That is neat. (I wish more explanation of why a name was chosen was available (on the Lab and elsewhere). It often helps understand the concept that bit better.
Did I use language or notation in some nonstandard way? LG, as I defined it, is a groupoid. So it has a nerve. Its morphisms are morphisms of G which relate their domain and codomain (that are free loops in G) by conjugation. The composition of such morphisms in LG is in fact given by multiplication in G, and so that is what goes into forming its nerve.
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