This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
As mentioned in §Cohomology, the acyclic models theorem generalizes the original proof of §Cohomology, ⁋Theorem 9, and can be applied not only to that theorem but also in numerous other situations. In this post we prove the acyclic models theorem and present several corollaries, including the proof of §Cohomology, ⁋Theorem 9.
Category with models
When developing homology theory we typically work with \(n\)-simplices, which allow us to probe arbitrary objects of \(\Top\). We codify 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 fruitful 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.
- 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 \(i>0\).
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The functor \(F_\bullet\) is free on \(\mathcal{M}\) if for each \(n\), the 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 \(\mathcal{M}\) consists of the standard \(n\)-simplices \(\Delta^n\). Then the functor \(C_\bullet:\Top \rightarrow \Ab\) assigning to each \(X\in \Top\) the chain complex \(C_\bullet(X)\) of singular \(n\)-simplices is both acyclic on \(\mathcal{M}\) and free on \(\mathcal{M}\).
- That \(C_\bullet\) is acyclic on \(\mathcal{M}\) follows from §Homology, ⁋Proposition 11. Note that the condition that \(F_\bullet\) be acyclic on \(\mathcal{M}\) does
not require the \(0\)-th homology of \(F_\bullet(X)\) to vanish. - That \(C_\bullet\) is free on \(\mathcal{M}\) is immediate, since each \(C_n(X)\) is the free abelian group generated by the 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 \((\mathcal{A},\mathcal{M})\) be a category with models, and let \(F_\bullet, G_\bullet:\mathcal{A}\rightarrow \Ch_{\geq0}(\lMod{A})\) be functors with \(F_\bullet\) free on \(\mathcal{M}\) and \(G_\bullet\) 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 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, let us 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 the following commutative diagram
defines the map \(F_0(X)\rightarrow H_0(G(X))\) in a canonical manner, and since \(p_G\) is surjective we obtain a lift \(F_0(X)\rightarrow G_0(X)\).
However, defining \(f_\bullet(X)\) in higher degrees presents a subtle 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 construct a lift in the diagram
but unlike the previous case, we require the newly defined \(f_n(X)\) to 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 condition it is not clear how \(f_n(X)\) should be defined.
To resolve this we invoke the hypothesis that \(G\) is acyclic on \(\mathcal{M}\). First, since the functor \(F_n\) is free, it suffices to define \(f_n\) on the models \(M\). For an arbitrary object \(X\), a free module \(F_n(X)\), and a generator \(u:M \rightarrow X\), the diagram
shows that the element of \(F_n(M)\) corresponding to \(\id_M\) becomes \(u\) in \(F_n(X)\), so we may send \(u\) to \((G_n(u)\circ f_n(M))(\id_M)\). Having thus reduced our attention to models, what remains is to lift the diagram
But now for any \(x_n\in F_n(M)\),
\[0=(f_{n-2}(M)\circ d_{n-1}^{F(M)}\circ d_n^{F(M)})(x_n)=(d_{n-1}^{G(M)}\circ f_{n-1}(M)\circ d_n^{F(M)})(x_m)\]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 therefore 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 \(f_\bullet(M)\). Different choices of \(y_n\) yield different lifts \(f_n\), and their difference defines a chain homotopy.
Applications of the acyclic models theorem
The acyclic models theorem is used first of all in proving the Künneth theorem discussed in the previous post. Consider the category \(\Top^2\) of pairs of topological spaces, and the two functors
\[C_\bullet(-\times -;A),\qquad C_\bullet(-;A)\otimes_A C_\bullet(-;A)\]from \(\Top^2\) to \(\Ch_{\geq 0}(\lMod{A})\). Taking the models \(\mathcal{M}\) to be the collection of
\[(\Delta^p, \Delta^q)\in\Top^2\]these functors are both free on \(\mathcal{M}\) and acyclic on \(\mathcal{M}\). The map
\[C_p(X;A)\times C_q(Y;A)\rightarrow C_{p+q}(X\times Y;A);\qquad (\sigma,\tau)\mapsto \sigma\times\tau\]is then an isomorphism on \(H_0\), and its lifting is the Eilenberg–Zilber map, while the lifting of its inverse is the Alexander–Whitney map.
As a similar example, consider the four functors from \(\Top^2\) to \(\Ch_{\geq 0}(\lMod{A})\)
\[(X,Y)\mapsto C_\bullet(X\times Y;A),\quad (X,Y)\mapsto C_\bullet(Y\times X;A),\quad (X,Y)\mapsto C_\bullet(X;A)\otimes_AC_\bullet(Y;A),\quad (X,Y)\mapsto C_\bullet(Y;A)\otimes_AC_\bullet(X;A)\]together with the obvious maps between them; lifting these via Theorem 3 yields a commutative diagram in \(\Ch_{\geq0}(\lMod{A})\)
References
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