clarification about magma cohomology

Stream: Equational

[Andy Jiang]

Hi,

I haven't been really following recently but I was kind of intrigued by the discussion on magma cohomology and had a basic question. So from my understanding there's some finite complex associated with a magma type and a linear model of that magma type of length equal to the number of variables. I'm curious what the relation to usual group cohomology is here, is it just with the group cohomology of the trivial module? Because I didn't understand how to put a general representation of a group's cohomology in this framework. Also does the magma cohomology have a cup product?

[Zoltan A. Kocsis (Z.A.K.)]

I'm not the right person to answer the question, but it might be useful context to whoever responds: have you read through the whole discussion thread (in Austin pairs and Ways to analyze) and the relevant writeup in the blueprint as well? In particular Remark 4 is relevant to the relation to group cohomology.

[Andy Jiang]

I haven't yet. But also skimming at the blueprint I didn't see the definition of the complex only one of the cohomology groups I don't know if I'm just not looking carefully though

[Bruno Le Floch]

The complex is described by Terry Tao here. The first cohomology group is the magma morphisms from the magma to the coefficient group. The second cohomology group involves the equation. There is no clear higher cohomology in general.

Reading the blueprint section, some observations.

• Interestingly, some equations like x = (y*x)*(z*y) have no linear model so no cohomology.
• It's odd that G is the magma and M is the group.
• A few expressions of the form Q(a,b) f(w1(…)) f(w2(…)) are typos for Q(a,b) f(w1(…),w2(…)).
• The cocycle equation for 543-magma cohomology (equation 543 is Tarski's equation characterizing abelian group subtraction) seems to match the usual abelian group cohomology. It would be interesting to do the same question for the Higman-Neumann equation 42323216 (characterizing group division).

[Terence Tao]

Thanks for the feedback! I have updated the blueprint chapter with some corrections and additional remarks (the PR at equational#996 should land on the web page in about twenty minutes).

I chose the notation $G$ for the base and $M$ for the linear model to be consistent with the usual notation for group cohomology, although I do agree that for magmas the notation would most naturally be the other way around.

Our current magma cohomology is indeed analogous to group cohomology with a trivial module. It's not clear to me whether it makes sense to consider non-trivial modules, given that the base $G$ is not required to be associative, so would not be expected to have many interesting representations.

Given that our cohomology stops at $H^2$, I doubt we have an interesting cup product. The most obvious candidate would be some operation $\wedge: H^1(G,M) \times H^1(G,M) \to H^2(G,M)$ that would come from some sort of tensor product operation $\otimes: C^1(G,M) \times C^1(G,M) \to C^2(G,M)$ that maps homomorphisms to cocycles, but I don't see how to construct such an operation in general.

[Adam Topaz]

The complex described in the other thread is missing the zero-th term. In the case of (inhomogeneous) cochains for group cohomology we have $C^0(G,M) = M$ and the boundary map sends $m$ to $\sigma \mapsto \sigma m - m$. This is relevant because, in the case of group cohomology $H^i(G,M)$ are the derived functors of $H^0(G,M) = M^G$. There are also more sophisticated descriptions of group cohomology from topology which we could look into as well.

[Andy Jiang]

Thanks for the replies!

Re: descriptions from topology one of the ways to think about group cohomology I believe is that there is a notion of a homotopy coherent associative group which has an underlying space instead of just a underlying set, for example the space of based loops on a nice topological space. Using this notion of group, any abelian group M gives rise to eilenberg maclane spaces K(M,n) which are associative groups in spaces. Then you can find the higher group cohomology as maps from G to K(M,n) as associative groups in spaces. So it could be possible to say here that this cohomology comes from the fact that you can make sense of a E-rule magma in spaces, and this group could be some form of map from G to BM as E-rule magmas. And the extension could be the fibre of this map. But likely because there's no natural notion of homotopy coherent E-magma it's unlikely to be able to extend it.

[Adam Topaz]

Yes, it's known that the Eilenberg-MacLane spaces $K(A,n)$ represent $H^n(-,A)$ for spaces, so that $H^n(X,A) = [X, K(A,n)]$ where $[-,-]$ means continuous maps modulo homotopy. But for group cohomology you also need to do something to the group $G$, namely consider the cohomology of its classifying space $BG$. This is a space that comes equipped with a principal $G$-bundle $EG \to BG$ and it's universal in the sense that principal $G$-bundles on $X$ (for nice enough $X$) correspond to homotopy equivalences of maps $X \to BG$ by pulling back $EG$. Since $EG$ is contractible in this case, you can also observe that $BG$ is a $K(G,1)$. (Also all of this ignores the situation where $G$ acts nontrivially on $A$, which adds some further complications.)

[Andy Jiang]

So I was working in the situation of trivial action of G on M. In this case group cohomology is the cohomology of the constant sheaf B^nM on BG as you say. However, this is identified with maps from BG to B^n(M), which as connected spaces are equivalent to grouplike E1 algebras in spaces (what I was calling homotopy coherent group above) this is the same as maps of homotopy coherent groups from G to B^{n-1}M. I think it's this interpretation which generalizes to the magma case.

[Adam Topaz]

Okay, yes, I agree this could be written down as a definition. But my guess is that associativity (at least upto homotopy, as in the E_1-algebras case) plays a key role in the usual case, so I'm not sure whether this would give a meaningful answer in the case of arbitrary magmas. There is another approach using Andr\'e-Quillen cohomology mentioned briefly in a math overflow answer here: https://mathoverflow.net/questions/148510/is-there-a-cohomology-for-magmas I think this approach is probably the most promising

[Andy Jiang]

Well my feeling is that because the homotopy coherent associative algebra structure generalizes the associativity law we get the higher group cohomologies. For a general magma, we can try the same approach without extending the law to a coherent law (in a sense extending it trivially to a homotopy coherent law with no higher coherences) This will then only give three nontrivial homotopy groups, the magma hom, the group discussed in this forum, and also the cokernel of the map from C2 to Ck. Andre-Quillen cohomology I believe is another name for cotangent complex formalisms classifying deformations and indeed probably generalize here as that answer says, but I know little about this approach in this context so I can't say more about the Andre Quillen cohomology here.

[Bruno Le Floch]

If magma equation A implies magma equation B, we know that the (second) A-cohomology group of a magma with coefficients in some linear A-model is contained in the (second) B-cohomology group. What data is needed to construct all B-cohomologies for all consequences B of equation A?

If A is associativity, the (second) A-cohomology group is usual $H^2$ group cohomology, and I suspect that you need precisely all the group cohomology to characterize all second B-cohomology groups for B consequences of A. For instance, I suspect that the (second) B-cohomology for B the equation x*(y*(z*w)) = ((x*y)*z)*w is related to the third cohomology group in the usual sense. EDIT: No, third and higher group cohomology involve "more data" in the sense that cocycles are functions of more than two group elements. In addition, the particular consequence of associativity that I picked has the same cohomology as associativity when considering magmas that are groups (basically because we can set one of the variables to the identity). OTOH, the x*(y*x) = (x*y)*x cohomology (say) of a group is not obviously fixed by its second group cohomology.

[Bruno Le Floch]

For central groupoids $G$, here is a natural-looking notion of $H^n(G,M)$ cohomology groups, $n\geq 1$.
Central groupoids obey equation 168, which I write $y=(x\diamond y)\diamond (y\diamond z)$, with a relabelling that orders arguments of cocycles conveniently below.
Linear models $(M,*)$ have the form $M=S\times S$ with

$$(s_1,s_2)*(s_3,s_4) = (s_2,s_3) = (s_1,s_2)*0 + 0*(s_3,s_4)$$

where $0=(0,0)\in M$. One has $0*((s_1,s_2)*0)=(0,s_2)$ and $(0*(s_1,s_2))*0=(s_1,0)$, and $0*(0*(s_1,s_2))=((s_1,s_2)*0)*0=0$.

Here is a chain complex: $C^n(G,M)$ is the group of functions from $G^n$ to $M$, and the differential $d^{n+1}\colon C^n(G,M)\to C^{n+1}(G,M)$ is defined by

$$(d^{(n+1)}f)(x_0,x_1,\dots,x_n) = f(x_0,\dots,x_{n-1})*0 + 0*f(x_1,\dots,x_n) + (-1)^n f(x_0\diamond x_1,x_1\diamond x_2,\dots,x_{n-1}\diamond x_n) .$$

One checks

$$(d^{(n+1)}d^nf)(x_0,x_1,\dots,x_n) = (0*f(x_1,\dots,x_{n-1}))*0 + ((-1)^{n-1} + (-1)^n) f(x_0\diamond x_1,\dots,x_{n-2}\diamond x_{n-1})*0 + 0*(f(x_1,\dots,x_{n-1})*0) + ((-1)^{n-1} + (-1)^n) 0*f(x_1\diamond x_2,\dots,x_{n-1}\diamond x_n) + (-1)^{2n-1} f\bigl((x_0\diamond x_1)\diamond(x_1\diamond x_2),\dots\bigr) = 0 .$$

Explicitly, the first few differentials are $d^{(1)}=0$ and

$$\begin{aligned} d^{(2)}f(x,y) & = f(x)*0 + 0*f(y) - f(x\diamond y) , \\ d^{(3)}f(x,y,z) & = f(x,y)*0 + 0*f(y,z) + f(x\diamond y,y\diamond z) , \\ d^{(4)}f(x,y,z,w) & = f(x,y,z)*0 + 0*f(y,z,w) - f(x\diamond y,y\diamond z,z\diamond w) . \end{aligned}$$

The differential $d^{(3)}$ comes from the equation itself, $y=(x\diamond y)\diamond(y\diamond z)$.
The differential $d^{(4)}$ comes from a coherence condition

$$y\diamond x = ((w\diamond y)\diamond (y\diamond x)) \diamond x = ((w\diamond y)\diamond (y\diamond x)) \diamond ((y\diamond x)\diamond (x\diamond z)) = y\diamond x$$

where we apply equation 168 to $y$, $x$, and reversed equation 168 to $y\diamond x$ (hence the sign in $d^{(4)}$).
I guessed the higher differentials by recognizing a pattern.

[Bruno Le Floch]

Even for groups, magma cohomology gives something new! For some group $G$ and coefficient group $M$, we can have $H^2_{4435}(G,M)$ large and all group cohomologies $H^k(G,M)$ trivial (for the same coefficient group). I could be wrong.

We know that the second group cohomology (with coefficients in a trivial module) is captured by magma cohomology for the associativity equation. Associativity has consequences such as (for instance) equation 4435 $x*(y*x)=(x*y)*x$. Consider the corresponding cohomology $H^2_{4435}(G,M)$.

Let me focus on $(G,+)$ abelian, but keep the coefficient group $M$ general. The cocyle equation is $f(y,x)+f(x,y+x)=f(x+y,x)+f(x,y)$. Equivalently, for $g(x,y)=f(x,y)-f(y,x)$, we find $g(x,y)=g(x,kx+y)$ and (by antisymmetry) $g(x,y)=g(x+ky,y)$ for any $k$, so

$$g(ax+by,cx+dy) = (-1)^{\det(a,b;c,d)} g(x,y) , \qquad \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{GL}(2,\mathbb{Z}) ,$$

with additionally $g(x,x)=0$ (in characteristic $2$ that is not a consequence of $\mathrm{GL}(2, \mathbb{Z})$ covariance). When $x,y$ generate a subgroup of $G$ with a single generator, I think this imposes $g(x,y)=0$: basically apply Euclid's algorithm, then note that $g(0,y) = g(y,y) = 0$. But when $x,y$ generate a two-generator subgroup of $G$, that group is not changed by the $\mathrm{GL}(2,\mathbb{Z})$ action so that $g(x,y)$ is never forced to be equal to some $g(w,w)=0$.

Concretely, consider $G=(\mathbb{Z}/3\mathbb{Z})^2$. Pairs $(x,y)=((x_1,x_2),(y_1,y_2))\in G^2$ have three orbits under $\mathrm{SL}(2,\mathbb{Z})$, characterized by the determinant of $(\begin{smallmatrix}x_1&x_2\\y_1&y_2\end{smallmatrix})$, which is $0$ (for $33$ pairs) or $\pm 1$ (for $24$ pairs each). The two latter orbits are exchanged by reflections in $\mathrm{GL}(2,\mathbb{Z})$. One can take the function $g$ to map the zero orbit to zero, and the other two orbits to two opposite values, so

$$g((x_1,x_2),(y_1,y_2)) = (x_1y_2-x_2y_1 \mod 3) m \in M ,$$

for some element $m\in M$, with the understanding that $(x_1y_2-x_2y_1 \mod 3)\in\{-1,0,1\}$ and then we do not impose that $M$ has a particular characteristic. To get an actual cocycle, we just need to complete this assignment of $g$ into an assignment of $f$: pick $f(x,x)$ arbitrarily, and for each two-element subset $\{x,y\}\subset G$, pick one of $f(x,y)$ or $f(y,x)$, the other is then fixed. Altogether, that's $|M|^{37}$ choices (choice of $m$, then one choice for each of the 36 pairs). This is much more than the number of coboundaries, which is at most $|M|^{9}$. So the cohomology group $H^2_{4435}(G,M)$ has at least $|M|^{28}$ elements.

On the other hand, when $M$ has characteristic coprime with $3$, my understanding is that the usual cohomology groups are trivial e.g., a similar calculation from Wikipedia.

[Andy Jiang]

Ah that's amazing, so it seems there should be a notion of a homotopy coherent central groupoid--could be worth adding to further directions maybe?

[Bruno Le Floch]

Thanks. I think it's worth exploring slightly more before deciding what to write. For a general equation, one can consider all of its coherence properties (not sure what the correct terminology is). I suspect that for associativity and for the central groupoid equation all coherence properties are consequences of a single one: pentagon identity for associativity, and the identity I used for central groupoids. For a more general equation you could have more than one "independent" coherence equation, so that the coboundary map would take values in something more complicated than $C^4(G,M)$.

[Andy Jiang]

I could be misunderstanding what you're saying here but I would not say the higher coherences of associativity are consequences of the lower ones. This sort of phenomenon happens when the space you impose the A infinity structure (or E1 structure--this just means homotopy coherent monoid) is truncated in some way, e.g for categories with hom-sets (as opposed to infinity categories). But for example it is possible to have spaces in which the lower coherence exists but the higher ones do not (i guess it's related to so called An algebras) I would expect just based on intuition the same for the central groupoid

[Bruno Le Floch]

Sorry I wasn't clear because I haven't yet learned properly about A infinity structures etc. I think what I meant is that in an A infinity structure you only have one "generator" with each arity, and likewise in the "CentralGroupoid infinity" structure (I guess), but for a generic equation you could have more than one generator at each arity (or more precisely at each order in the construction of cohomology groups). If you just focus on "lowest" coherence of the equation you may have multiple analogues of the pentagon equation (which may by the way involve different numbers of variables, hence have different arities, but that's a minor point).

[Bruno Le Floch]

Is there a way to automatically search for syzygies to define higher cohomologies? Namely, given an equation such as equation 168, $x=(y\diamond x)\diamond (x\diamond z)$, find the coherence condition

$$y\diamond x = ((w\diamond y)\diamond (y\diamond x)) \diamond x = ((w\diamond y)\diamond (y\diamond x)) \diamond ((y\diamond x)\diamond (x\diamond z)) = y\diamond x$$

that I used to define the third cohomology group. Given the ongoing discussion around equation 677, $x=y\diamond(x\diamond((y\diamond x)\diamond y))$, I'm especially interested in finding syzygies for that equation, and by hand I am failing to do so. It might be related to the rarity of models of this equation despite it being only a two-variable equation: the equations for different values of $x,y$ seem very independent of each other.