Kan fibration explained

In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category 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.

For various kinds of fibrations for simplicial sets, see Fibration of simplicial sets.

Definitions

Definition of the standard n-simplex

For each n ≥ 0, recall that the standard

-simplex,

\Deltaⁿ

, is the representable simplicial set

\Deltaⁿ⁽ⁱ⁾=Hom_\Delta([i],[n])

Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard

-simplex: the convex subspace of

Rⁿ⁺¹

consisting of all points

(t_0,...,t_n)

such that the coordinates are non-negative and sum to 1.

Definition of a horn

For each k ≤ n, this has a subcomplex

	n
Λ
	k

, the k-th horn inside

\Deltaⁿ

, 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

\Delta^n-1 → \Deltaⁿ

corresponding to all the other faces of

\Deltaⁿ

.^[1] Horns of the form

	2
Λ
	k

sitting inside

\Delta²

look like the black V at the top of the adjacent image. If

is a simplicial set, then maps

	n
Λ
	k

\toX

correspond to collections of

(n-1)

-simplices satisfying a compatibility condition, one for each

0\leqk\leqn-1

. Explicitly, this condition can be written as follows. Write the

(n-1)

-simplices as a list

(s_0,...,s_k-1,s_k+1,...,s_n)

and require that

d_is_j=d_j-1s_i

for all

i<j

with

i,j ≠ k

These conditions are satisfied for the

(n-1)

-simplices of

	n
Λ
	k

sitting inside

\Deltaⁿ

Definition of a Kan fibration

A map of simplicial sets

f:X → Y

is a Kan fibration if, for any

n\ge1

and

0\lek\len

, and for any maps

	n
s:Λ
	k →

and

y:\Delta^{n →}Y

such that

f\circs=y\circi

(where

is the inclusion of

	n
Λ
	k

\Deltaⁿ

), there exists a map

x:\Deltaⁿ → X

such that

s=x\circi

and

y=f\circx

. Stated this way, the definition is very similar to that of fibrations in topology (see also homotopy lifting property), whence the name "fibration".

Technical remarks

Using the correspondence between

-simplices of a simplicial set

and morphisms

\Deltaⁿ\toX

(a consequence of the Yoneda lemma), this definition can be written in terms of simplices. The image of the map

fs:

	n
Λ
	k

\toY

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

x:\Delta^n\toX

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.^[2]

Kan complexes defined from Kan fibrations

A simplicial set

is called a Kan complex if the map from

X\to\{*\}

, the one-point simplicial set, is a Kan fibration. In the model category 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

\alpha:

	n
Λ
	k

\toX

from a horn has an extension to

\Deltaⁿ

, meaning there is a lift

\tilde{\alpha}:\Deltaⁿ\toX

such that

\alpha=\tilde{\alpha}\circ\iota

for the inclusion map

\iota:

	n
Λ
	k

\hookrightarrow\Deltaⁿ

, 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, called the singular functor^[3] ^{pg 7}

S:Top\tosSets

.

Given a space

, define a singular

-simplex of X to be a continuous map from the standard topological

-simplex (as described above) to

f:\Delta_n\toX

Taking the set of these maps for all non-negative

gives a graded set,

S(X)=\coprod_nS_n(X)

.To make this into a simplicial set, define face maps

d_i:S_n(X)\toS_n-1(X)

(d_if)(t_0,...,t_n-1)=f(t_0,...,t_i-1,0,t_i,...,t_n-1)

and degeneracy maps

s_i:S_n(X)\toS_n+1(X)

(s_if)(t_0,...,t_n+1)=f(t_0,...,t_i-1,t_i+t_i+1,t_i+2,...,t_n+1)

.Since the union of any

n+1

faces of

\Delta_n+1

is a strong deformation retract of

\Delta_n+1

, any continuous function defined on these faces can be extended to

\Delta_n+1

, which shows that

S(X)

is a Kan complex.^[4]

Relation with geometric realization

It is worth noting the singular functor is right adjoint to the geometric realization functor

| ⋅ |:sSets\toTop

giving the isomorphism

Hom_Top(|X|,Y)\congHom_sSets(X,S(Y))

See also: Milnor's theorem on Kan complexes.

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, its geometric realization is homotopy equivalent to a product of Eilenberg-Maclane spaces

\prod_iK(A_i,n_i)

In particular, this includes classifying spaces. So the spaces

S¹\simeqK(Z,1)

CP^infty\simeqK(Z,2)

, and the infinite lens spaces

	infty
L
	q

\simeqK(Z/q,2)

are correspond to Kan complexes of some simplicial set. In fact, this set can be constructed explicitly using the Dold–Kan correspondence 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

l{G}

. This is defined as the geometric realization of the simplicial set

[\Delta^op,l{G}]

and is typically denoted

Bl{G}

. We could have also replaced

l{G}

with an infinity groupoid. It is conjectured that the homotopy category of geometric realizations of infinity groupoids is equivalent to the homotopy category of homotopy types. This is called the homotopy hypothesis.

Mapping space

Let

be an ∞-category. Then for each object

x,y

, let

\operatorname{Map}_X(x,y)

be the fiber of

(s,t):\underline{\operatorname{Hom}}(\Delta^1,X)\toX²

over the point

(x,y)

. Then

\operatorname{Map}_X(x,y)

is a Kan complex.

Postnikov section

Let X be a Kan complex. Then the n-th Postnikov section

P_nX

is a simplicial set such that

P_nX_m

is the coequalizer of

X_m\rightrightarrows\operatorname{cosk}_nX_m

.^[5] Then the following can be verified directly:

P_nX

is a Kan complex and

X\toP_nX

is a Kan fibration.

The induced map

\pi_iX\to\pi_iP_nX

is an isomorphism for

i=0

and for each

0<i\len

and choice of a base point on X.

\pi_iX=0

for each

i>n

and choice of a base point on X.

For a simplicial set X, we then let

P_nX=P_n\operatorname{Ex}^inftyX

where

\operatorname{Ex}^infty

is an Ex∞ functor.

Non-example: standard n-simplex

It turns out the standard

-simplex

\Deltaⁿ

is not a Kan complex^[6] ^{pg 38}. The construction of a counter example in general can be found by looking at a low dimensional example, say

\Delta¹

. Taking the map

	2
Λ
	0

\to\Delta¹

sending

\begin{matrix} [0,2]\mapsto[0,0]&[0,1]\mapsto[0,1] \end{matrix}

gives a counter example since it cannot be extended to a map

\Delta²\to\Delta¹

because the maps have to be order preserving. If there was a map, it would have to send

\begin{align} 0\mapsto0\\ 1\mapsto1\\ 2\mapsto0 \end{align}

but this isn't a map of simplicial sets.

Categorical properties

Simplicial enrichment and function complexes

For simplicial sets

X,Y

there is an associated simplicial set called the function complex

bf{Hom}(X,Y)

, where the simplices are defined as

bf{Hom}_n(X,Y)=Hom_sSets(X x \Delta^n,Y)

and for an ordinal map

\theta:[m]\to[n]

there is an induced map

\theta^*:bf{Hom}(X,Y)_n\tobf{Hom}(X,Y)_m

(since the first factor of Hom is contravariant) defined by sending a map

f:X x \Deltaⁿ\toY

to the composition

X x \Delta^m\xrightarrow{1 x \theta}X x \Deltaⁿ\xrightarrow{f}Y

Exponential law

This complex has the following exponential law of simplicial sets

ev_*:Hom_sSets(K,bf{Hom}(X,Y))\toHom_sSets(X x K,Y)

which sends a map

f:K\tobf{Hom}(X,Y)

to the composite map

X x K\xrightarrow{1 x g}X x bf{Hom}(X,Y)\xrightarrow{ev}Y

where

ev(x,f)=f(x,\iota_n)

for

\iota_n\inHom_{\Delta([n],[n])}

lifted to the n-simplex

\Deltaⁿ

Kan fibrations and pull-backs

Given a (Kan) fibration

p:X\toY

and an inclusion of simplicial sets

i:K\hookrightarrowL

, there is a fibration ^{pg 21}

*,p
bf{Hom}(L,X)\xrightarrow{(i
*)}bf{Hom}(K,X) x

_bf{Hom(K,Y)}bf{Hom}(L,Y)

(where

bf{Hom}

is in the function complex in the category of simplicial sets) induced from the commutative diagram

\begin{matrix} bf{Hom}(L,X)&\xrightarrow{p_*}&bf{Hom}(L,Y)\\ i^*\downarrow&&\downarrowi^*\\ bf{Hom}(K,X)&\xrightarrow{p_*}&bf{Hom}(K,Y) \end{matrix}

where

i^*

is the pull-back map given by pre-composition and

p_*

is the pushforward map given by post-composition. In particular, the previous fibration implies

p_{*:bf{Hom}(L,X)\to}bf{Hom}(L,Y)

and

i^{*:bf{Hom}(L,Y)\to}bf{Hom}(K,Y)

are fibrations.

The above is a consequence of a theorem of Gabriel and Zisman.

Homotopy groups of Kan complexes

The homotopy groups of a fibrant simplicial set may be defined combinatorially, using horns, in a way that agrees with the homotopy groups of the topological space which realizes it. For a Kan complex

and a vertex

x:\Delta⁰\toX

, as a set

\pi_n(X,x)

is defined as the set of maps

\alpha:\Deltaⁿ\toX

of simplicial sets fitting into a certain commutative diagram:

\pi_n(X,x)= \left\{ \alpha:\Deltaⁿ\toX: \begin{matrix} \Deltaⁿ&\overset{\alpha}{\to}&X\\ \uparrow&&\uparrowx\\ \partial\Deltaⁿ&\to&\Delta^{0
\end{matrix}}\right\}

Notice the fact

\partial\Deltaⁿ

is mapped to a point is equivalent to the definition of the sphere

Sⁿ

as the quotient

Bⁿ/\partialBⁿ

for the standard unit ball

Bⁿ=\{x\inRⁿ:||x||_eu\leq1\}

Defining the group structure requires a little more work. Essentially, given two maps

\alpha,\beta:\Deltaⁿ\toX

there is an associated

(n+1)

-simplice

\omega:\Deltaⁿ⁺¹\toX

such that

	n
d
	n\omega:\Delta

\toX

gives their addition. This map is well-defined up to simplicial homotopy classes of maps, giving the group structure. Moreover, the groups

\pi_n(X,x)

are Abelian for

n\geq2

. For

\pi_0(X)

, it is defined as the homotopy classes

[x]

of vertex maps

x:\Delta⁰\toX

Homotopy groups of simplicial sets

Using model categories, any simplicial set

has a fibrant replacement

\hat{X}

which is homotopy equivalent to

in the homotopy category of simplicial sets. Then, the homotopy groups of

can be defined as

\pi_n(X,x):=\pi_{n(\hat{X},\hat{x})}

where

\hat{x}

is a lift of

x:\Delta⁰\toX

\hat{X}

. These fibrant replacements can be thought of a topological analogue of resolutions of a chain complex (such as a projective resolution or a flat resolution).

Kan

Kan complexes themselves form a weak Kan complex called Kan. Namely, first consider the category K where objects are Kan complexes and morphisms maps of simplicial sets. As a category of presheaves has internal Hom, each hom-set in K has a structure of a simplicial set; in short, K is a simplicial category. The homotopy coherent nerve of it

bf{Kan}:=N^hc(K)

is then a weak Kan complex (∞-category).^[7] In view of homotopy hypothesis, it is often taken as the ∞-category of spaces = ∞-groupoids and is also denoted as

or some other variants.

Footnotes

See Goerss and Jardine, page 7
May uses this simplicial definition; see
Book: Goerss. Paul G.. Simplicial Homotopy Theory. Jardine. John F.. 2009. Birkhäuser Basel. 978-3-0346-0188-7. 837507571.
See May, page 3
§ 5.6. in Book: Jardine, John F.. Rick Jardine
. Rick Jardine. Local homotopy theory. Springer Monographs in Mathematics. Springer-Verlag. New York. 2015. 3309296. 10.1007/978-1-4939-2300-7. 978-1-4939-2299-4.
Friedman. Greg. 2016-10-03. An elementary illustrated introduction to simplicial sets. math.AT. 0809.4221.
Web site: Construction 5.5.1.1 (The ∞-Category of Spaces). Keradon: an online resource for homotopy-coherent mathematics. Jacob. Lurie. Jacob Lurie.

References

10.1216/RMJ-2012-42-2-353. 0809.4221 . Survey Article: An elementary illustrated introduction to simplicial sets . 2012 . Friedman . Greg . Rocky Mountain Journal of Mathematics . 42 . 2 .
Book: Goerss . Paul G. . Jardine . John F. . Rick Jardine. Simplicial Homotopy Theory . Birkhäuser Basel . Basel . 1999 . 978-3-0348-9737-2 . 10.1007/978-3-0348-8707-6 . 1711612 .
Book: May, J. Peter . J. Peter May . Simplicial objects in algebraic topology . . Chicago, IL. Chicago Lectures in Mathematics. 1992. 1967. 0-226-51180-4. 1206474.
Pierre Gabriel, Michel Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) https://people.math.rochester.edu/faculty/doug/otherpapers/GZ.pdf
Book: Cisinski, Denis-Charles . Denis-Charles Cisinski . Higher Categories and Homotopical Algebra . 2023. . 978-1108473200 . en .
10.1016/j.indag.2010.12.004 . Algebraic models for higher categories . 2011 . Nikolaus . Thomas . Indagationes Mathematicae . 21 . 1–2 . 52–75 . 1003.1342 .