Acyclic Models Theorem

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

As mentioned in §Cohomology, the acyclic models theorem generalizes the original proof of §Cohomology, ⁋Theorem 9 (Eilenberg-Zilber) in a systematic way, and it can be applied not only to the proof of §Cohomology, ⁋Theorem 9 (Eilenberg-Zilber) but also to a variety of other situations. In this post, we prove the acyclic models theorem and present several corollaries, including the proof of §Cohomology, ⁋Theorem 9 (Eilenberg-Zilber).

Category with models

When developing homology theory, we typically work with \(n\)-simplices, which allow us to probe arbitrary objects of \(\Top\). We formalize this as follows.

Definition 1 A category with models is a pair \((\mathcal{A},\mathcal{M})\) consisting of a category \(\mathcal{A}\) and a collection \(\mathcal{M}\) of objects of \(\mathcal{A}\). The objects belonging to \(\mathcal{M}\) are called models.

This definition is not particularly substantial on its own. We now introduce the following.

Definition 2 Let \((\mathcal{A},\mathcal{M})\) be a category with models, and let \(F_\bullet:\mathcal{A}\rightarrow \Ch_{\geq0}(\lMod{A})\) be a covariant functor.

1. The functor \(F_\bullet\) is acyclic on \(\mathcal{M}\) if for every \(M\in\mathcal{M}\), we have \(H_i(F(M))=0\) for all \(i>0\).

2. The functor \(F_\bullet\) is free on \(\mathcal{M}\) if for each \(n\) there exists a family of models \((M_j)_{j\in J_n}\) such that the natural isomorphism

   \[F_n(-)\cong \bigoplus_{j\in J_n}A[\Hom_\mathcal{A}(M_j,-)]\]

   holds. Here \(A[S]\) denotes the free \(A\)-module with basis the set \(S\), and each family \((M_j)_{j\in J_n}\) may contain the same model multiple times.

For example, consider the category with models \((\Top, \mathcal{M})\) where \(\mathcal{M}\) is the collection of standard \(n\)-simplices \(\Delta^n\). Then the functor \(C_\bullet:\Top \rightarrow \Ch_{\geq0}(\Ab)\) assigning to each \(X\in \Top\) the singular chain complex \(C_\bullet(X)\) is both acyclic on \(\mathcal{M}\) and free on \(\mathcal{M}\).

• That \(C_\bullet\) is acyclic on \(\mathcal{M}\) follows because each model \(\Delta^n\) is a convex set, hence contractible to a point, and the cone operator induced by the straight-line contraction gives a direct contraction of \(C_\bullet(\Delta^n)\) in positive degrees; this can be viewed as a generalization of §Homology, ⁋Proposition 11. Note that the condition that \(F_\bullet\) is acyclic on \(\mathcal{M}\) does not require the \(0\)th homology of \(F_\bullet(X)\) to vanish.
• That \(C_\bullet\) is free on \(\mathcal{M}\) follows because each \(C_n(X)\) is the free abelian group with basis precisely the singular \(n\)-simplices \(\Delta^n \rightarrow X\), that is, \(C_n(X)=\mathbb{Z}[\Hom_\Top(\Delta^n,X)]\). In this case, the family of models taken for each \(n\) consists of the single model \(\Delta^n\).

Acyclic models theorem

The main theorem of this post is the following.

Theorem 3 (Acyclic models theorem) Let \((\mathcal{A},\mathcal{M})\) be a category with models, and let \(F_\bullet, G_\bullet:\mathcal{A}\rightarrow \Ch_{\geq0}(\lMod{A})\) be two functors such that \(F_\bullet\) is free on \(\mathcal{M}\) and \(G_\bullet\) is acyclic on \(\mathcal{M}\). Then for any natural transformation

\[f(-)_0:H_0(F(-)) \Rightarrow H_0(G(-))\]

between the two functors

\[H_0(F(-)),H_0(G(-)): \mathcal{A}\rightarrow \lMod{A}\]

there exists a natural transformation

\[f_\bullet(-):F_\bullet(-) \rightarrow G_\bullet(-)\]

with \(H_0(f)=f(-)_0\), and such a natural transformation \(f\) is unique up to natural chain homotopy.

That is, starting from \(f(X)_0: H_0(F(X))\rightarrow H_0(G(X))\) defined at the homology level, we must construct a chain map \(f_\bullet(X):F_\bullet(X)\rightarrow G_\bullet(X)\). To this end, we first define the \(0\)th component \(f_0(X)\). Since \(F_0(X)\) is free, this amounts to specifying the image of each \(u:M\rightarrow X\). Now, by the following commutative diagram

the composition \(F_0(X)\rightarrow H_0(F(X))\rightarrow H_0(G(X))\) is given, and since \(p_G\) is surjective we obtain a lift \(F_0(X)\rightarrow G_0(X)\). However, if we choose a lift separately for each \(X\), there is no guarantee that these choices are natural with respect to one another; the role of the models \(\mathcal{M}\) is to resolve this. Specifically, for each model \(M\) we only choose the image of the element of \(F_0(M)\) corresponding to \(\id_M\), that is, \(f_0(M)(\id_M)\), and then for the remaining generators \(u:M\rightarrow X\) we define \(f_0(X)(u):=(G_0(u)\circ f_0(M))(\id_M)\). The \(f_0\) defined in this way is natural by the functoriality of \(G_0\), and the fact that it is still a lift of \(f(X)_0\) follows from the naturality of \(f(-)_0\).

However, defining \(f_\bullet(X)\) in higher degrees presents an additional difficulty. Suppose inductively that the components up to \(f_{n-1}(X)\) have been defined, and let us define \(f_n(X)\). That is, we must define the lift in the following diagram

but unlike the situation above, we must require that the newly defined \(f_n(X)\) satisfy the commutativity condition

\[d_n^{G(X)}\circ f_n(X)=f_{n-1}(X)\circ d_n^{F(X)}\]

Moreover, even without this commutativity condition, it is not clear how \(f_n(X)\) should be defined.

To resolve this, we use the hypothesis that \(G\) is acyclic on \(\mathcal{M}\). First, since the functor \(F_n\) is free, we know that it suffices to define \(f_n\) on the models \(M\). For any object \(X\), the free module \(F_n(X)\), and a generator \(u:M \rightarrow X\), the following diagram

shows that the element of \(F_n(M)\) corresponding to \(\id_M\) becomes \(u\) in \(F_n(X)\), so we need only send \(u\) to \((G_n(u)\circ f_n(M))(\id_M)\). Having shifted our attention to models, what remains is to lift the preceding diagram

Now if \(n\geq2\), then for any \(x_n\in F_n(M)\),

\[0=(f_{n-2}(M)\circ d_{n-1}^{F(M)}\circ d_n^{F(M)})(x_n)=(d_{n-1}^{G(M)}\circ f_{n-1}(M)\circ d_n^{F(M)})(x_n)\]

so by the assumption that \(G\) is acyclic on \(\mathcal{M}\),

\[f_{n-1}(d_n^{F(M)}(x_n))\in \ker d_{n-1}^{G(M)}=\im d_n^{G(M)}\]

and thus we can find \(y_n\) satisfying \(d_n^{G(M)}(y_n)=f_{n-1}(d_n^{F(M)}(x_n))\), from which we obtain the \(n\)th component of the chain map \(f_\bullet(M)\).

In the case \(n=1\), for any \(x_1\in F_1(M)\), the element \(d_1^{F(M)}(x_1)\) is a boundary in \(F(M)\) and hence vanishes in \(H_0(F(M))\); since \(f_0\) was chosen to lift \(f(M)_0\), the class determined by \(f_0(d_1^{F(M)}(x_1))\) in \(H_0(G(M))\) also vanishes. That is, \(f_0(d_1^{F(M)}(x_1))\in \ker p_G=\im d_1^{G(M)}\), and thus we can find \(y_1\) in the same manner. Of course, the \(f_\bullet\) obtained in this way depends on the choice of \(y_n\) and is therefore not unique, but one can verify that the difference between two choices is absorbed by a natural chain homotopy.

Applications of the acyclic models theorem

The acyclic models theorem is used first of all in proving §Cohomology, ⁋Corollary 10 (Künneth). Consider the category \(\Top^2\) of pairs of topological spaces, and the two functors to \(\Ch_{\geq 0}(\lMod{A})\)

\[C_\bullet(-\times -;A),\qquad C_\bullet(-;A)\otimes_A C_\bullet(-;A)\]

If we take the models \(\mathcal{M}\) to be the collection of

\[(\Delta^p, \Delta^q)\in\Top^2\]

then both functors are free on \(\mathcal{M}\) and acyclic on \(\mathcal{M}\). The natural transformation determined by sending \([\sigma]\otimes[\tau]\) to \([(\sigma,\tau)]\) for \(0\)-simplices \(\sigma,\tau\)

\[H_0(C_\bullet(X;A)\otimes_AC_\bullet(Y;A))\cong H_0(X;A)\otimes_AH_0(Y;A)\rightarrow H_0(X\times Y;A)\]

is an isomorphism because \(\pi_0(X\times Y)=\pi_0(X)\times\pi_0(Y)\); its lift is the Eilenberg–Zilber map, and the lift of its inverse is the Alexander–Whitney map. However, one must be careful that at the chain level we cannot write \(\sigma\times\tau\) directly. This is because the product of \(\sigma:\Delta^p\rightarrow X\) and \(\tau:\Delta^q\rightarrow Y\) is defined on the prism \(\Delta^p\times\Delta^q\), not on \(\Delta^{p+q}\), and decomposing this prism into simplices is precisely what the Eilenberg–Zilber map accomplishes.

As a similar example, consider the four functors from \(\Top^2\) to \(\Ch_{\geq 0}(\lMod{A})\)

\[(X,Y)\mapsto C_\bullet(X\times Y;A),\quad (X,Y)\mapsto C_\bullet(Y\times X;A),\quad (X,Y)\mapsto C_\bullet(X;A)\otimes_AC_\bullet(Y;A),\quad (X,Y)\mapsto C_\bullet(Y;A)\otimes_AC_\bullet(X;A)\]

The obvious maps between them, when lifted via Theorem 3 (Acyclic models theorem), yield a diagram in \(\Ch_{\geq0}(\lMod{A})\) that commutes up to natural chain homotopy

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References

The method of acyclic models

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