In mathematics, Kan complexes and Kan fibrations are part of the theory of
simplicial set
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
s. Kan fibrations are the fibrations of the standard
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 abstra ...
structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the
fibrant objects in this model category. The name is in honor of
Daniel Kan
Daniel Marinus Kan (or simply Dan Kan) (August 4, 1927 – August 4, 2013) was a Dutch mathematician working in category theory and homotopy theory. He was a prolific contributor to both fields for six decades, having authored or coauthored sever ...
.
Definitions
Definition of the standard n-simplex
For each ''n'' ≥ 0, recall that the
standard -simplex,
, is the representable simplicial set
:
Applying the
geometric realization functor to this simplicial set gives a space homeomorphic to the
topological standard -simplex: the convex subspace of ℝ
n+1 consisting of all points
such that the coordinates are non-negative and sum to 1.
Definition of a horn
For each ''k'' ≤ ''n'', this has a subcomplex
, the ''k''-th horn inside
, corresponding to the boundary of the ''n''-simplex, with the ''k''-th face removed. This may be formally defined in various ways, as for instance the union of the images of the ''n'' maps
corresponding to all the other faces of
. Horns of the form
sitting inside
look like the black V at the top of the adjacent image. If
is a simplicial set, then maps
:
correspond to collections of
-simplices satisfying a compatibility condition, one for each
. Explicitly, this condition can be written as follows. Write the
-simplices as a list
and require that
:
for all
with
.
These conditions are satisfied for the
-simplices of
sitting inside
.
Definition of a Kan fibration
A map of simplicial sets
is a Kan fibration if, for any
and
, and for any maps
and
such that
(where
is the inclusion of
in
), there exists a map
such that
and
. Stated this way, the definition is
very similar to that of
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all ma ...
s in
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
(see also
homotopy lifting property In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from ...
), whence the name "fibration".
Technical remarks
Using the correspondence between
-simplices of a simplicial set
and morphisms
(a consequence of the
Yoneda lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (vie ...
), this definition can be written in terms of simplices. The image of the map
can be thought of as a horn as described above. Asking that
factors through
corresponds to requiring that there is an
-simplex in
whose faces make up the horn from
(together with one other face). Then the required map
corresponds to a simplex in
whose faces include the horn from
. The diagram to the right is an example in two dimensions. Since the black V in the lower diagram is filled in by the blue
-simplex, if the black V above maps down to it then the striped blue
-simplex has to exist, along with the dotted blue
-simplex, mapping down in the obvious way.
Kan complexes defined from Kan fibrations
A simplicial set
is called a Kan complex if the map from
, the one-point simplicial set, is a Kan fibration. In 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 abstra ...
for simplicial sets,
is the terminal object and so a Kan complex is exactly the same as a
fibrant object. Equivalently, this could be stated as: if every map
from a horn has an extension to
, meaning there is a lift
such that
for the inclusion map
, then
is a Kan complex. Conversely, every Kan complex has this property, hence it gives a simple technical condition for a Kan complex.
Examples
Simplicial sets from singular homology
An important example comes from the construction of
singular simplices used to define
singular homology
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''- ...
, called the singular functor
pg 7.
Given a space
, define a singular
-simplex of X to be a continuous map from the standard topological
-simplex (as described above) to
,
:
Taking the set of these maps for all non-negative
gives a graded set,
:
.
To make this into a simplicial set, define face maps
by
:
and degeneracy maps
by
:
.
Since the union of any
faces of
is a strong
deformation retract
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformat ...
of
, any continuous function defined on these faces can be extended to
, which shows that
is a Kan complex.
Relation with geometric realization
It is worth noting the singular functor is
right adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
to the
geometric realization functorgiving the isomorphism
Simplicial sets underlying simplicial groups
It can be shown that the simplicial set underlying a
simplicial group is always fibrant
pg 12. In particular, for a
simplicial abelian group In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial grou ...
, its geometric realization is homotopy equivalent to a product of Eilenberg-Maclane spaces
In particular, this includes
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
s. So the spaces
,
, and the infinite lens spaces
are correspond to Kan complexes of some simplicial set. In fact, this set can be constructed explicitly using the
Dold–Kan correspondence
In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the ...
of a chain complex and taking the underlying simplicial set of the simplicial abelian group.
Geometric realizations of small groupoids
Another important source of examples are the simplicial sets associated to a small groupoid
. This is defined as the geometric realization of the simplicial set