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 broad manner, and it can be used in various situations, not only when proving §Cohomology, ⁋Theorem 9 (Eilenberg-Zilber). In this post, we prove the acyclic models theorem and present some corollaries, including the proof of §Cohomology, ⁋Theorem 9 (Eilenberg-Zilber).

Category with models

When developing homology theory, we usually employ $n$-simplices, which help us examine arbitrary objects of $\Top$. We can formulate this as the following definition.

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 very substantial on its own. We now define the following.

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

The functor $F_\bullet$ is said to be acyclic on $\mathcal{M}$ if for each $M\in\mathcal{M}$, we have $H_i(F(M))=0$ for all $i>0$.
The functor $F_\bullet$ is said to be free on $\mathcal{M}$ if for each $n$, the following natural isomorphism
\[F_n(-)\cong \bigoplus_{M\in \mathcal{M}}\mathbb{Z}\Hom_\mathcal{A}(M,-)\]
holds.

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

That $C_\bullet$ is acyclic on $\mathcal{M}$ follows from the fact that each model $\Delta^n$ is a convex set, hence contractible to a point, and the straight-line contraction induces a cone operator that directly contracts $C_\bullet(\Delta^n)$ for $i>0$; this can be regarded as a generalization of §Homology, ⁋Proposition 11. Note here that the condition that the functor $F_\bullet$ is acyclic on $\mathcal{M}$ does not require that the $0$th homology of $F_\bullet(X)$ be zero.
That $C_\bullet$ is free on $\mathcal{M}$ is immediate because each $C_n(X)$ is precisely the free abelian group generated by maps $\Delta^n \rightarrow X$, i.e., $C_n(X)=\mathbb{Z}\Hom_\Top(\Delta^n,X)$.

Acyclic models theorem

The main theorem of this post is the following.

Theorem 3 (Acyclic models theorem) Let a category with models $(\mathcal{A},\mathcal{M})$ and two functors $F_\bullet, G_\bullet:\mathcal{A}\rightarrow \Ch_{\geq0}(\lMod{A})$ be given, and suppose $F_\bullet$ 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

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

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 do this, we first define the $0$th component $f_0(X)$ of $f_\bullet(X)$. Since $F_0(X)$ is free, this amounts to defining where each $u:M\rightarrow X$ is sent. On the other hand, by the following commutative diagram

$lifting$

$F_0(X)\rightarrow H_0(G(X))$ is defined in an obvious way, and since $p_G$ is surjective, we can define a lifting $F_0(X)\rightarrow G_0(X)$ from this.

However, defining $f_\bullet(X)$ in higher degrees presents a slight 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 need to define the lifting in the following diagram

$lifting_general$

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

To resolve this, we use the condition 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 $X$ and free module $F_n(X)$, and for a generator $u:M \rightarrow X$, using the following diagram

$reduction_to_models$

the element of $F_n(M)$ corresponding to $\id_M$ becomes $u$ in $F_n(X)$, and then we simply send $u$ to $(G_n(u)\circ f_n(M))(\id_M)$. Having shifted our attention to the models, what we need to do is lift the preceding diagram

$lifting_reduced$

But now for any $x_n\in F_n(M)$, since

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

by the assumption that $G$ is acyclic on $\mathcal{M}$ we have

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

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 can construct the $n$th component of the chain map $f_\bullet(M)$. Different choices of $y_n$ give different lifts $f_n$, and the difference between these defines a chain homotopy.

Applications of the acyclic models theorem

The acyclic models theorem is first used in proving the Künneth theorem examined in the previous post. Consider the category $\Top^2$ of pairs of topological spaces, and the two functors from it to $\Ch_{\geq 0}(\lMod{A})$

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

Now if we take the model $\mathcal{M}$ to be the collection of

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

then these are all free on $\mathcal{M}$ and acyclic on $\mathcal{M}$. The following function

\[C_p(X;A)\times C_q(Y;A)\rightarrow C_{p+q}(X\times Y;A);\qquad (\sigma,\tau)\mapsto \sigma\times\tau\]

is an isomorphism on $H_0$, and then the lifting of this function becomes the Eilenberg–Zilber map, while the lifting of the inverse of this function becomes the Alexander–Whitney map.

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

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

then we can consider the obvious maps between these, and lifting these using Theorem 3 (Acyclic models theorem) yields the commutative diagram in $\Ch_{\geq0}(\lMod{A})$

$flip_map$

References

The method of acyclic models

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Category with models

Acyclic models theorem

Applications of the acyclic models theorem

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