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In [[algebraic topology]], a discipline within [[mathematics]], the '''acyclic models theorem''' can be used to show that two [[homology theories]] are [[isomorphic]]. The [[theorem]] was developed by topologists [[Samuel Eilenberg]] and [[Saunders MacLane]]. They discovered that, when topologists were writing proofs to establish equivalence of various homology theories, there were numerous similarities in the processes.  Eilenberg and MacLane then discovered the theorem to generalize this process.
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It can be used to prove the [[Eilenberg–Zilber theorem]].
 
==Statement of the theorem==
Let <math>\mathcal{K}</math> be an arbitrary [[category (mathematics)|category]] and <math>\mathcal{C}(R)</math> be the category of chain complexes of <math>R</math>-[[Module (mathematics)|module]]s. Let <math>F,V : \mathcal{K} \to \mathcal{C}(R)</math> be [[covariant functor]]s such that:
* <math> F_i = V_i = 0 </math> for <math> i < 0</math>.
* There are <math>\mathcal{M}_k \subseteq \mathcal{K}</math> for <math>k  \ge 0</math> such that <math>F_k</math> has a basis in <math> \mathcal{M}_k </math>, so <math>F</math> is a [[free functor]].
* <math>V</math> is <math>k</math>- and <math>(k+1)</math>-acyclic at these models, which means that <math>H_k(V(M)) = 0</math> for all <math>k>0</math> and all <math>M \in \mathcal{M}_k \cup \mathcal{M}_{k+1}</math>.
 
Then the following assertions hold:
* Every [[natural transformation]] <math>\varphi : H_0(F) \to H_0(V)</math> is induced by a natural chain map <math>f : F \to V</math>.
* If <math>\varphi,\psi: H_0(F)\to H_0(V)</math> are natural transformations, <math>f,g: F\to V</math> are natural chain maps as before and <math>\varphi^{M}=\psi^{M}</math> for all models <math>M\in\mathcal{M}_0</math>, then there is a natural chain homotopy between <math>f</math> and <math>g</math>.
* In particular the chain map <math> f</math> is unique up to natural [[chain homotopy]].<ref> {{Citation | last1=Dold | first1=Albrecht | title=Lectures on Algebraic Topology | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=A Series of Comprehensive Studies in Mathematics | isbn= 3-540-10369-4 | edition=2nd | year=1980 | volume=200}}</ref>
 
== Generalizations ==
 
=== Projective and acyclic complexes ===
 
What is above is one of the earliest versions of the theorem. Another
version is the one that says that if <math>K</math> is a complex of
projectives in an [[abelian category]] and <math>L</math> is an acyclic
complex in that category, then any map <math>K_0 \to
L_0</math> extends to a chain map <math>K\to L</math>, unique up to
homotopy.
 
This specializes almost to the above theorem if one uses the functor category <math>\mathcal{C}(R)^\mathcal{K}</math> as the abelian category. Free functors are projective objects in that category. The morphisms in the functor category are natural transformations, so the constructed chain maps and homotopies are all natural. The difference is that in the above version, <math>V</math> being acyclic is a stronger assumption than being acyclic only at certain objects.
 
On the other hand, the above version almost implies this version by letting <math>\mathcal{K}</math> a category with only one object. Then the free functor <math>F</math> is basically just free (and hence projective) module. <math>V</math> being acyclic at the models (there is only one) means nothing else than that the complex <math>V</math> is acyclic.
 
=== Acyclic classes ===
 
Then there is the grand theorem that unifies them all.  Let <math>
\mathcal{A}</math> be an abelian category (for example <math>\mathcal{C}(R)</math> or <math>\mathcal{C}(R)^\mathcal{K}</math>). A class <math>\Gamma</math> of chain complexes over <math>\mathcal{A}</math> will be called an '''acyclic class''' provided:
* The 0 complex is in <math>\Gamma</math>.
* The complex <math>C</math> belongs to <math>\Gamma</math> if and only if the suspension of <math>C</math> does.
* If the complexes <math>K</math> and <math>L</math> are homotopic and <math>K \in\Gamma</math>, then <math>L\in\Gamma</math>.
* Every complex in <math>\Gamma</math> is acyclic.
* If <math>D</math> is a double complex, all of whose rows are in <math>\Gamma</math>, then the total complex of <math>D</math> belongs to <math>\Gamma</math>.
 
There are three natural examples of acyclic classes, although doubtless
others exist. The first is that of homotopy contractible complexes.
The second is that of acyclic complexes. In functor categories (e.g. the
category of all functors from topological spaces to abelian groups),
there is a class of complexes that are contractible on each object, but
where the contractions might not be given by natural transformations.
Another example is again in functor categories but this time the complexes are acyclic only at certain objects.
 
Let <math>\Sigma</math> denote the class of chain maps between complexes
whose mapping cone belongs to <math>\Gamma</math>. Although
<math>\Sigma</math> does not necessarily have a calculus of either right
or left fractions, it has weaker properties of having homotopy classes
of both left and right fractions that permit forming the class
<math>\Sigma^{-1} C</math> gotten by inverting the arrows in
<math>\Sigma</math>.{{Citation needed|date=February 2010}}
 
Let <math>G</math> be an augmented endofunctor on <math>C</math>,
meaning there is given a natural transformation
<math>\epsilon:G\to Id</math> (the identity functor on <math>C</math>). We say that the chain complex <math>K</math> is <math>G</math>-''presentable'' if for each <math>n</math>, the chain
complex
:<math>\cdots K_nG^{m+1}\to K_nG^{m}\to \cdots \to K_n</math>
belongs to <math>\Gamma</math>. The boundary operator is given by
:<math>\sum (-1)^i K_nG^i\epsilon G^{m-i}:K_nG^{m+1}\to K_nG^m</math>.
We say that the chain complex functor <math>L</math> is
<math>G</math>-''acyclic'' if the augmented chain complex
<math>L\to H_0(L)\to 0</math> belongs to <math>\Gamma</math>.
 
'''Theorem'''. ''Let <math>\Gamma</math> be an acyclic class and
<math>\Sigma</math> the corresponding class of arrows in the category of
chain complexes. Suppose that <math>K</math> is <math>G</math>-presentable and <math>L</math> is <math>G</math>-acyclic.
Then any natural transformation <math>f_0:H_0(K)\to H_0(L)</math>
extends, in the category <math>\Sigma^{-1}(C)</math> to a natural
transformation of chain functors <math>f:K\to L</math> and this is
unique in <math>\Sigma^{-1}(C)</math> up to chain homotopies.
If we suppose, in addition, that <math>L</math> is <math>G</math>-presentable, that <math>K</math> is <math>G</math>-acyclic, and that <math>f_0</math> is an isomorphism, then <math>f</math> is homotopy equivalence.
 
== Example ==
Here is an example of this last theorem in action. Let <math>X</math>
be the category of triangulable spaces and <math>C</math> be the
category of abelian group valued functors on <math>X</math>. Let
<math>K</math> be the singular chain complex functor and <math>L</math>
be the simplicial chain complex functor. Let <math>E: X\to
X</math> be the functor that assigns to each space <math>X</math> the
space <math>\sum_{n\ge 0}\sum_{\textrm{Hom}(\Delta_n,X)}\Delta_n</math>. Here, <math>\Delta_n</math> is the <math>n</math>-simplex and this functor assigns to <math>X</math> the sum of as many copies of each
<math>n</math>-simplex as there are maps <math>\Delta_n\to X</math>.
Then let <math>G</math> be defined by <math>G(C)=CE</math>. There is an
obvious augmentation <math>EX\to X</math> and this induces one on
<math>G</math>. It can be shown that both <math>K</math> and
<math>L</math> are both <math>G</math>-presentable and
<math>G</math>-acyclic (the proof that <math>L</math> is not entirely
straigtforward and uses a detour through simplicial subdivision, which
can also be handled using the above theorem). The class <math>\Gamma</math> is the class of homology equivalences. It is rather
obvious that <math>H_0(K)\simeq H_0(L)</math> and so we conclude that
singular and simplicial homology are isomorphic on <math>X</math>.
 
There are many other examples in both algebra and topology, some of
which are described in M. Barr, Acyclic Models. AMS, 2002.
 
==References==
<references/>
 
 
* Schon, R. Acyclic models and excision. _Proc. Amer. Math. Soc._ 59~(1) (1976) 167--168.
 
 
[[Category:Homological algebra]]
[[Category:Theorems in algebraic topology]]
{{DEFAULTSORT:Acyclic Model}}

Revision as of 01:06, 26 February 2014

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