Acyclic Models Theorem

As mentioned in §Cohomology, the acyclic models theorem extends the original proof of §Cohomology, ⁋Theorem 9 (Eilenberg-Zilber) in a general way, and can be used not only to prove §Cohomology, ⁋Theorem 9 (Eilenberg-Zilber) but also in various situations. In this post, we prove the acyclic models theorem and introduce several corollaries, including the proof of §Cohomology, ⁋Theorem 9 (Eilenberg-Zilber).

Category with models

When developing homology theory, we usually work with \(n\)-simplices, which help us examine arbitrary objects of \(\Top\). We can formulate this into a definition as follows.

Definition 1 A category with models means a pair consisting of a category \(\mathcal{A}\) and a collection of objects of \(\mathcal{A}\) denoted \(\mathcal{M}\), written \((\mathcal{A},\mathcal{M})\). Here, we call the objects belonging to \(\mathcal{M}\) models.

This definition has little substance on its own. We now define 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 each \(M\in\mathcal{M}\), 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, with the standard \(n\)-simplices \(\Delta^n\) forming the collection \(\mathcal{M}\) of models, consider the category with models \((\Top, \mathcal{M})\). Then, assigning to each \(X\in \Top\) the chain complex of singular \(n\)-simplices \(C_\bullet(X)\), the functor \(C_\bullet:\Top \rightarrow \Ch_{\geq0}(\Ab)\) 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 and hence contractible to a point, and the cone operator induced by that straight-line contraction directly contracts \(C_\bullet(\Delta^n)\) for \(i>0\); this can be viewed as a generalization of §Homology, ⁋Proposition 11. Note that the condition that the functor \(F_\bullet\) is acyclic on \(\mathcal{M}\) does not require that for \(F_\bullet(X)\), the \(0\)th homology be \(0\).
• 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 the two functors

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

whenever any natural transformation between them

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

is given, there exists a suitable natural transformation

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

such that \(H_0(f)=f(-)_0\), 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, for \(f_\bullet(X)\), let us first define its \(0\)th component \(f_0(X)\). Since \(F_0(X)\) is free, this is equivalent to defining where each \(u:M\rightarrow X\) is sent. Meanwhile, by the following commutative diagram

the composite \(F_0(X)\rightarrow H_0(F(X))\rightarrow H_0(G(X))\) is given, and since \(p_G\) is surjective, we obtain a lifting \(F_0(X)\rightarrow G_0(X)\) from this. However, if we choose a lifting separately for each \(X\), there is no guarantee that these satisfy naturality with each other, and resolving this is the role of the models \(\mathcal{M}\). That is, for each model \(M\), we choose only where the element corresponding to \(\id_M\) in \(F_0(M)\) is sent, namely \(f_0(M)(\id_M)\), 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 from the functoriality of \(G_0\), and that this is still a lift of \(f(X)_0\) follows from the naturality of \(f(-)_0\).

However, to define \(f_\bullet(X)\) in higher degrees, there is a slight problem. 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 a lifting of the following diagram

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

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

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

To resolve this, we use the condition that \(G\) is acyclic on \(\mathcal{M}\). First, from the fact that the functor \(F_n\) is free, we know that it suffices to define \(f_n\) only on the models \(M\). This is because, for any object \(X\), free module \(F_n(X)\), and generator \(u:M \rightarrow X\), if we use the following diagram

the element corresponding to \(\id_M\) in \(F_n(M)\) becomes \(u\) in \(F_n(X)\), and we need only send \(u\) to \((G_n(u)\circ f_n(M))(\id_M)\). Now, once we shift our focus to the models, what we must do is lift the previous 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 from 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 therefore, we can find, satisfying \(d_n^{G(M)}(y_n)=f_{n-1}(d_n^{F(M)}(x_n))\), a \(y_n\), from which we can construct for the chain map \(f_\bullet(M)\) its \(n\)th component.

In the case of \(n=1\), for any \(x_1\in F_1(M)\), since \(d_1^{F(M)}(x_1)\) is a boundary in \(F(M)\), its class in \(H_0(F(M))\) is \(0\), and since \(f_0\) was chosen to lift \(f(M)_0\), the class determined by \(f_0(d_1^{F(M)}(x_1))\) in \(H_0(G(M))\) is also \(0\). 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 \(\Top^2\) consisting of pairs of topological spaces, and consider the two functors from this category to \(\Ch_{\geq 0}(\lMod{A})\)

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

Now, if we take the models \(\mathcal{M}\) to be the collection of

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

they are all free on \(\mathcal{M}\) and acyclic on \(\mathcal{M}\). Now, for \(0\)-simplices \(\sigma,\tau\), the natural transformation determined by the assignment sending \([\sigma]\otimes[\tau]\) to \([(\sigma,\tau)]\)

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

is an isomorphism since \(\pi_0(X\times Y)=\pi_0(X)\times\pi_0(Y)\), and then a lifting of this becomes the Eilenberg-Zilber map, and a lifting of its inverse becomes the Alexander-Whitney map. However, one should note 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 does.

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

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

we can consider the obvious maps between them, and lifting these using Theorem 3 (Acyclic models theorem) yields 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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