Acyclic Models


The theorem(s) of acyclic models are a rather powerful technique for proving that some functors defined on truncated chain complexes can be extended in higher dimensions, and that two such functors are homotopic, by proving it on a small class of “model” objects.

For some reason I only discovered this last year, and I always find myself forgetting the precise hypotheses and conclusion… Hopefully writing this blog post will fix them in my mind. My main reference will be:

Unless indicated otherwise, all the definitions, theorems… come from there.

The theorem

“Representable” functors

Let C\mathsf{C} be any category (for example topological spaces, simplicial set…). Let also MobC\mathcal{M} \subset \operatorname{ob} \mathsf{C} be a set of objects, called (acyclic) model objects (think about standard simplexes).

For a functor T:CAbT : \mathsf{C} \to \mathsf{Ab} to the category of abelian groups, let T~:CAb\widetilde{T} : \mathsf{C} \to \mathsf{Ab} be the new functor defined as follows:

There is also a natural transformation Φ:T~T\Phi : \widetilde{T} \to T given by ΦA(M,φ,m)=T(φ)(m)\Phi_A(M, \varphi, m) = T(\varphi)(m). Morally, this is similar to the construction where you take an abelian group and you want to view it as a quotient of a free object: you take the free abelian group generated by all elements of your group, and you quotient by the relation ggggg \cdot g' \equiv gg'. Here we want to view all the T(A)T(A) as generated by the T(M)T(M).

The functor TT is then said to be representable (the terminology is unfortunate…) if Φ\Phi has a right inverse, i.e. there is a natural transformation Ψ:TT~\Psi : T \to \widetilde{T} such that ΦΨ\Phi\Psi is the identity of TT.

In other words, this means that for all ACA \in \mathsf{C}, T(A)T(A) has a “basis” (consisting of models objects) which is described by Ψ\Psi: ΨA:T(A)T~(A)\Psi_A : T(A) \to \widetilde{T}(A) sends xT(A)x \in T(A) to a sum α(Mα,φα,mα)\sum_{\alpha} (M_\alpha, \varphi_\alpha, m_\alpha) such that

x=αT(φα)(mα).x = \sum_\alpha T(\varphi_\alpha)(m_\alpha).

When a functor is representable, we will be able to check equalities on M\mathcal{M}, and equalities on C\mathsf{C} will follow.

Chain complexes

Now suppose we are given a functor K:CChK : \mathsf{C} \to \mathsf{Ch} from our category C\mathsf{C} to the category of nonnegatively graded chain complexes. We will denote q:Kq(A)Kq1(A)\partial_q : K_q(A) \to K_{q-1}(A) the differential.

If KK and LL are two such functors, then we can define maps KLK \to L in the obvious way, and we can also define truncated maps in degrees n\le n. Similarly given two maps, we can define homotopies between then, as well as truncated homotopies, which satisfy

q+1hq+hq1q=gqfq\partial_{q+1} h_q + h_{q-1} \partial_q = g_q - f_q

in degrees n\le n.

Theorem. Let K,L:CChK,L : \mathsf{C} \to \mathsf{Ch} be two functors and f:KLf : K \to L be a map truncated in degrees <q< q. If KqK_q is representable and Hq1(L(M))=0H_{q-1}(L(M)) = 0 for all MMM \in \mathcal{M}, then ff can be extended in dimension qq.

Proof. We want to define fq:KqLqf_q : K_q \to L_q such that qfqfq1q1\partial_q f_q f_{q-1} \partial_{q-1}. For all models MM and all mKq(M)m \in K_q(M), the element fq1(qm)f_{q-1}(\partial_q m) is a cycle in Lq1(M)L_{q-1}(M); but Lq1(M)L_{q-1}(M) is acyclic, thus we can choose some λ(m)Lq(M)\lambda(m) \in L_q(M) such that qλ(m)=fq1qm\partial_q \lambda(m) = f_{q-1} \partial_q m.

Now we let Λ:Kq~Lq\Lambda : \widetilde{K_q} \to L_q be defined by

ΛA(M,φ,m):=Lq(φ)(λ(m)).\Lambda_A(M, \varphi, m) := L_q(\varphi)(\lambda(m)).

It’s easy to check that this is a natural transformation, and that qΛ=fq1qΦ\partial_q \Lambda = f_{q-1} \partial_q \Phi (where Φ:Kq~Kq\Phi : \widetilde{K_q} \to K_q is the canonical morphism).

But since KqK_q was representable, we can let fq=ΛΨf_q = \Lambda \Psi where Ψ\Psi is the right inverse to Φ\Phi, and everything works out. ■

Do you see what we did (well, Eilenberg–MacLane did) here? It was easy to define an “fqf_q-like” map on the free stuff generated by the models, because we could just choose an arbitrary lift for each element without bothering to check if they were compatible with each other. The “representability” of KqK_q then provides a way to combine these all together. (This is also why we needed M\mathcal{M} to be a set – I don’t think it would have been possible to “choose” these lifts if it were a proper class…).

The proof of the following theorem is almost the same (exercise!).

Theorem. Let K,L:CChK,L : \mathsf{C} \to \mathsf{Ch} be two functors, f,g:KLf,g : K \to L two maps and h:fgh : f \simeq g a homotopy truncated in degrees <q< q. If KqK_q is representable and Hq(L(M))=0H_q(L(M)) = 0 for all MMM \in \mathcal{M}, then hh can be extended in dimension qq.

These two theorems combine to give:

Theorem. Let K,L:CChK,L : \mathsf{C} \to \mathsf{Ch} be two functors, and f:KLf : K \to L be a map truncated in degrees <q< q. If KnK_n is representable for all nqn \ge q and Hn(L(M))=0H_n(L(M)) = 0 for all models and for all nq1n \ge q - 1, then ff admits a unique (up to homotopy) extension in all degrees.

Example: action of a monoid on a group

Let WW be a monoid and Λ=Z[W]\Lambda = \mathbb{Z}[W] be its algebra. We may view WW as a category C\mathsf{C} with a unique object and homC(,)=W\hom_{\mathsf{C}}(*,*) = W. We set M=obC={}\mathcal{M} = \operatorname{ob} \mathsf{C} = \{*\}.

A functor T:CAbT : \mathsf{C} \to \mathsf{Ab} is the same thing as a group equipped with a left action of WW, i.e. a left Λ\Lambda-module ; the functor T~\tilde{T} is the free abelian group on the set of pairs W×GW \times G, with WW acting as w(w,g):=(ww,g)w \cdot (w',g) := (ww', g). In other words, it’s the free Λ\Lambda-module on GG. The functor TT is then representable iff GG is projective as a Λ\Lambda-module.

Normalization of singular chains

Let S:TopChS_* : \mathsf{Top} \to \mathsf{Ch} be the classical “singular chains” functor. The abelian group Sn(X)S_n(X) is free on maps ΔnX\Delta^n \to X, where Δn\Delta^n is the standard nn-simplex:

Δn={(x0,,xn)[0,1]n+1xi=1},\Delta^n = \{ (x_0, \dots, x_n) \in [0,1]^{n+1} \mid \sum x_i = 1 \},

and n:Sn(X)Sn1(X)\partial_n : S_n(X) \to S_{n-1}(X) is the usual differential n=i(1)idi\partial_n = \sum_i (-1)^i d_i. We consider the augmented version with S1(X)=ZS_{-1}(X) = \mathbb{Z} and 0(x)=1\partial_0(x) = 1 for all xX=S0(X)x \in X = S_0(X).

Now Δ\Delta^\bullet is in fact a cosimplicial space, and so we get degeneracy maps sj:Sn(X)Sn+1(X)s_j : S_n(X) \to S_{n+1}(X). For a simplex σ:ΔnX\sigma : \Delta^n \to X, its degeneracy is given by:

sj(σ)(x0,,xn+1)=σ(x0,,xj+xj+1,,xn+1).s_j(\sigma)(x_0, \dots, x_{n+1}) = \sigma(x_0, \dots, x_j + x_{j+1}, \dots, x_{n+1}).

These, together with the did_i, satisfy the usual simplicial identities. One can then normalize the singular chains functor, either by letting

Sˉn+1(X)=Sn+1(X)/im(sn:Sn(X)Sn+1(X))\bar{S}_{n+1}(X) = S_{n+1}(X) / \operatorname{im}(s_n : S_n(X) \to S_{n+1}(X))

(the complex “normalized at the top”) or by letting

SN(X)=Sn(X)/j=0n1im(sj)S^N(X) = S_n(X) / \bigcup_{j=0}^{n-1} \operatorname{im}(s_j)

(the “normalized complex”).

In the following theorem, SS' is either SNS^N or Sˉ\bar{S}.

Theorem. Let f:SSf : S \to S' be the quotient map. Then there is a map g:SSg : S' \to S and homotopies gfidSgf \simeq \operatorname{id}_S and fgidSfg \simeq \operatorname{id}_{S'}.

The proof directly uses acyclic models. The set of models can be taken to be the set of standard simplexes {Δnn0}\{ \Delta^n \mid n \ge 0 \} (Eilenberg–MacLane use the class of all contractible spaces, but it’s not a set and their proof only uses standard simplexes).

In degree (1)(-1), all three complexes are equal (which gives the base case for the induction). One needs to know that the homology of Δn\Delta^n is trivial (both the normalized and the unnormalized one), which can be proven directly; then one needs to prove that the three functors SS, SNS^N and SS' are representable, which is almost immediate from the definition.

The interesting thing (IMO) is that one can go through the proof and see that this yields a completely explicit homotopy equivalence between the unnormalized complex and the normalized ones.

Eilenberg–Zilber theorem

This part is not from the article of Eilenberg–MacLane (but it’s nevertheless completely classical, see e.g. MacLane’s book Homology, chapter VIII.8).

Let MM and NN be two simplicial modules over some ring RR. One can produce two chain complexes out this: either take the two Moore complexes MM_*, NN_* and then take their tensor product, or take the diagonal of the product (M×N)n=Mn×Nn(M \times N)_n = M_n \times N_n and then take the Moore complexes. We will denote the two complexes respectively as MNM_* \otimes N_* and (M×N)(M \times N)_*. Of course this yields two functors (sModR)2Ch(s\mathsf{Mod}_R)^2 \to \mathsf{Ch}, to which we will apply the previous techniques.

The two complexes are both equal to M0N0M_0 \otimes N_0 is degree zero. This gives the base case for the induction (we can just take the map to be the identity). One can then choose as models the simplicial modules given by ΔnR\Delta^n \otimes R, which can easily be proven to be acyclic. The adjunctions

homsModR(ΔnR,M)homsSet(Δn,M)Mn\hom_{s\mathsf{Mod}_R}(\Delta^n \otimes R, M_\bullet) \cong \hom_{s\mathsf{Set}}(\Delta^n, M_\bullet) \cong M_n

can then be used to prove that both functors are representable. (I’m omitting a lot of computations here! Though most of it is straightforward.) The acyclic models technique then yields the equivalence:

(M×N)MN.(M \times N)_* \simeq M_* \otimes N_*.

Again, what’s really interesting is that both maps (and both homotopies!) can be described completely explicitly once you make the right choices (you need to go back to the proof of the first theorem to know what choices I’m talking about). The map f:(M×N)MNf : (M \times N)_* \to M_* \otimes N_* is known as the Alexander–Whitney map, and it is given by (for aAna \in A_n, bBnb \in B_n):

f(a×b)=i=0ndˉniad0ib,f(a \times b) = \sum_{i = 0}^n \bar{d}^{n-i}a \otimes d_0^i b,

where d0:BkBk1d_0 : B_k \to B_{k-1} is the 00th face and dˉ=dk:AkAk1\bar{d} = d_k : A_k \to A_{k-1} is the “last” face.

Conversely, g:MN(M×N)g : M_* \otimes N_* \to (M \times N)_* is known as the Eilenberg–Zilber map, given by (for aApa \in A_p, bBqb \in B_q):

g(ab)=(μ,ν)Shp,q±sνa×sμb.g(a \otimes b) = \sum_{(\mu, \nu) \in \mathrm{Sh}_{p,q}} \pm s*{\nu} a \times s*{\mu} b.

The sum runs over all (p,q)(p,q)-shuffles, with (for p+q=np+q = n):

Shp,q={(μ,ν){1,,n}p×{1,,n}qμ(1)<<μ(p),ν(1)<<ν(q),μ(i)ν(j)}\mathrm{Sh}_{p,q} = \{ (\mu, \nu) \in \{1,\dots,n\}^p \times \{1,\dots,n\}^q \mid \mu(1) < \dots < \mu(p), \nu(1) < \dots < \nu(q), \mu(i) \neq \nu(j) \}

and where sμ=sμ(p)sμ(1)s_{\mu} = s_{\mu(p)} \circ \dots \circ s_{\mu(1)}, and sν=sν(q)sν(1)s_{\nu} = s_{\nu(q)} \circ \dots \circ s_{\nu(1)}. (The reader is encouraged to see explicitly what this all means in small cases, say (p,q)=(1,2)(p,q) = (1,2)).

This provides a completely algebraic proof of the “topological” Eilenberg–Zilber theorem which states that, given two topological spaces XX and YY, one has S(X×Y)S(X)S(Y)S_*(X \times Y) \simeq S_*(X) \otimes S_*(Y), which in turns is a key ingredient in the Künneth theorem (which now becomes a purely algebraic statement about the homology of the tensor product of two chain complexes).