In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, a discipline within
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the acyclic models theorem can be used to show that two
homology theories are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. The
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
was developed by topologists
Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.
Early life and education
He was born in Warsaw, Kingdom of Poland to a ...
and
Saunders MacLane
Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg.
Early life and education
Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville, ...
. 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
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 appeare ...
; this leads to the idea of the
model category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstrac ...
.
Statement of the theorem
Let
be an arbitrary
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
and
be the category of chain complexes of
-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s over some ring
. Let
be
covariant functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
s such that:
*
for
.
* There are
for
such that
has a basis in
, so
is a
free functor
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between elem ...
.
*
is
- and
-acyclic at these models, which means that
for all
and all
.
Then the following assertions hold:
[ ]
* Every
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 (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
induces 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 In homological algebra in mathematics, the homotopy category ''K(A)'' of chain complexes in an additive category ''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain co ...
.
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
is a complex of
projectives in an
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 prototypical example of an abelian category is the category of ab ...
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 a 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
There is a grand theorem that unifies both of the above.
[M. Barr,]
Acyclic Models
(1999).[M. Barr, ''Acyclic Models'' (2002) CRM monograph 17, American Mathematical Society .] Let
be an abelian category (for example,
or
). A class
of chain complexes over
will be called an acyclic class provided that:
* 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 Mapping cone may refer to one of the following two different but related concepts in mathematics:
* Mapping cone (topology)
* Mapping cone (homological algebra)
In homological algebra, the mapping cone is a construction on a map of chain complexes ...
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
gotten by inverting the arrows in
.
Let
be an augmented endofunctor on
, meaning there is given a natural transformation
(the identity functor on
). We say that the chain complex
is
-''presentable'' if for each
, the chain complex
:
belongs to
. The boundary operator is given by
:
.
We say that the chain complex functor
is
-''acyclic'' if the augmented chain complex
belongs to
.
Theorem. ''Let
be an acyclic class and
the corresponding class of arrows in the category of chain complexes. Suppose that
is
-presentable and
is
-acyclic. Then any natural transformation
extends, in the category
to a natural transformation of chain functors
and this is
unique in
up to chain homotopies. If we suppose, in addition, that
is
-presentable, that
is
-acyclic, and that
is an isomorphism, then
is homotopy equivalence.
Example
Here is an example of this last theorem in action. Let
be the
category of triangulable spaces and
be the category of abelian group valued functors on
. Let
be the
singular chain complex
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...
functor and
be the
simplicial chain complex functor. Let
be the functor that assigns to each space
the space
:
.
Here,
is the
-simplex and this functor assigns to
the sum of as many copies of each
-simplex as there are maps
. Then let
be defined by
. There is an obvious augmentation
and this induces one on
. It can be shown that both
and
are both
-presentable and
-acyclic (the proof that
is presentable and acyclic is not entirely straightforward and uses a detour through simplicial subdivision, which can also be handled using the above theorem). The class
is the class of homology equivalences. It is rather obvious that
and so we conclude that singular and simplicial homology are isomorphic on
.
There are many other examples in both algebra and topology, some of which are described in
References
* Schon, R. "Acyclic models and excision." ''Proc. Amer. Math. Soc.'' 59(1) (1976) pp.167--168.
Homological algebra
Theorems in algebraic topology
{{DEFAULTSORT:Acyclic Model