Acyclic models theorem - AbsoluteAstronomy.com

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, a discipline within mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the acyclic models theorem can be used to show that two homology theories are isomorphic. The theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

was developed by topologists Samuel Eilenberg

Samuel Eilenberg

Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...

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.

It can be used to prove the Eilenberg–Zilber theorem

Eilenberg–Zilber theorem

In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space X \times Y and those of the spaces X and Y. The theorem first appeared in a 1953 paper in the American Journal of...

Statement of the theorem

Let

be an arbitrary category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

and

be the category of chain complexes of

-module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

s. Let

be covariant functors such that:

for .
There are for such that has a basis in , so is a free functor.
is - and -acyclic at these models, which means that for all and all .

Then the following assertions hold:

Every natural transformation
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

is induced by a natural chain map .
If are natural transformations, are natural chain maps as before and for all models , then there is a natural chain homotopy between and .
In particular the chain map is unique up to natural chain homotopy.

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

is a complex of
projectives in an abelian category

Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

and

is an acyclic
complex in that category, then any map

extends to a chain map

, unique up to
homotopy.

This specializes almost to the above theorem if one uses the functor category

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,

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

a category with only one object. Then the free functor

is basically just free (and hence projective) module.

being acyclic at the models (there is only one) means nothing else than that the complex

is acyclic.

Acyclic classes

Then there is the grand theorem that unifies them all. Let

be an abelian category (for example

). A class

of chain complexes over

will be called an acyclic class provided:

The 0 complex is in .
The complex belongs to if and only if the suspension of does.
If the complexes and are homotopic and , then .
Every complex in is acyclic.
If is a double complex, all of whose rows are in , then the total complex of belongs to .

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

denote the class of chain maps between complexes
whose mapping cone belongs to

. Although

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