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classical model structure on simplicial sets

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Idea

The classical model structure on simplicial sets or Kan-Quillen model structure , sSet QuillensSet_{Quillen} (Quillen 67, II.3) is a model category structure on the category sSet of simplicial sets which represents the standard classical homotopy theory.

Its weak equivalences are the weak homotopy equivalences (isomorphisms on simplicial homotopy groups), its fibrations are the Kan fibrations and its cofibrations are the monomorphisms (degreewise injections).

The singular simplicial complex/geometric realization adjunction constitutes a Quillen equivalence between sSet QuillensSet_{Quillen} and Top QuillenTop_{Quillen}, the classical model structure on topological spaces. This is sometimes called part of the statement of the homotopy hypothesis for Kan complexes. In the language of (∞,1)-category theory this means that sSet QuillensSet_{Quillen} and Top QuillenTop_{Quillen} both are presentations of the (∞,1)-category ∞Grpd of ∞-groupoids.

There are also other model structures on sSet itself, see at model structure on simplicial sets for more. This entry here focuses on just the standard classical model structure.

Background on combinatoral topology

This section reviews basics of the theory of simplicial sets (the modern version of the original “combinatorial topology”) necessary to define, verify and analyse the classical model category structure on simplicial sets, below. See also at simplicial homotopy theory.

Simplicial sets

The concept of simplicial sets is secretly well familiar already in basic algebraic topology: it reflects just the abstract structure carried by the singular simplicial complexes of topological spaces, as in the definition of singular homology and singular cohomology.

Conversely, every simplicial set may be geometrically realized as a topological space. These two adjoint operations turn out to exhibit the homotopy theory of simplicial sets as being equivalent (Quillen equivalent) to the homotopy theory of topological spaces. For some purposes, working in simplicial homotopy theory is preferable over working with topological homotopy theory.

Definition

For nn \in \mathbb{N}, the topological n-simplex is, up to homeomorphism, the topological space whose underlying set is the subset

Δ n{x n+1| i=0 nx i=1andi.x i0} n+1 \Delta^n \coloneqq \{ \vec x \in \mathbb{R}^{n+1} | \sum_{i = 0 }^n x_i = 1 \; and \; \forall i . x_i \geq 0 \} \subset \mathbb{R}^{n+1}

of the Cartesian space n+1\mathbb{R}^{n+1}, and whose topology is the subspace topology induces from the canonical topology in n+1\mathbb{R}^{n+1}.

Example

For n=0n = 0 this is the point, Δ 0=*\Delta^0 = *.

For n=1n = 1 this is the standard interval object Δ 1=[0,1]\Delta^1 = [0,1].

For n=2n = 2 this is the filled triangle.

For n=3n = 3 this is the filled tetrahedron.

Definition

For nn \in \mathbb{N}, n1\n \geq 1 and 0kn0 \leq k \leq n, the kkth (n1)(n-1)-face (inclusion) of the topological nn-simplex, def. 1, is the subspace inclusion

δ k:Δ n1Δ n \delta_k : \Delta^{n-1} \hookrightarrow \Delta^n

induced under the coordinate presentation of def. 1, by the inclusion

n n+1 \mathbb{R}^n \hookrightarrow \mathbb{R}^{n+1}

which “omits” the kkth canonical coordinate:

(x 0,,x n1)(x 0,,x k1,0,x k,,x n). (x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_n) \,.
Example

The inclusion

δ 0:Δ 0Δ 1 \delta_0 : \Delta^0 \to \Delta^1

is the inclusion

{1}[0,1] \{1\} \hookrightarrow [0,1]

of the “right” end of the standard interval. The other inclusion

δ 1:Δ 0Δ 1 \delta_1 : \Delta^0 \to \Delta^1

is that of the “left” end {0}[0,1]\{0\} \hookrightarrow [0,1].

(graphics taken from Friedman 08)

Definition

For nn \in \mathbb{N} and 0k<n0 \leq k \lt n the kkth degenerate (n)(n)-simplex (projection) is the surjective map

σ k:Δ nΔ n1 \sigma_k : \Delta^{n} \to \Delta^{n-1}

induced under the barycentric coordinates of def. 1 under the surjection

n+1 n \mathbb{R}^{n+1} \to \mathbb{R}^n

which sends

(x 0,,x n)(x 0,,x k+x k+1,,x n). (x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n) \,.
Definition

For XX \in Top and nn \in \mathbb{N}, a singular nn-simplex in XX is a continuous map

σ:Δ nX \sigma : \Delta^n \to X

from the topological nn-simplex, def. 1, to XX.

Write

(SingX) nHom Top(Δ n,X) (Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X)

for the set of singular nn-simplices of XX.

(graphics taken from Friedman 08)

The sets (SingX) (Sing X)_\bullet here are closely related by an interlocking system of maps that make them form what is called a simplicial set, and as such the collection of these sets of singular simplices is called the singular simplicial complex of XX. We discuss the definition of simplicial sets now and then come back to this below in def. 8.

Since the topological nn-simplices Δ n\Delta^n from def. 1 sit inside each other by the face inclusions of def. 2

δ k:Δ n1Δ n \delta_k : \Delta^{n-1} \to \Delta^{n}

and project onto each other by the degeneracy maps, def. 3

σ k:Δ n+1Δ n \sigma_k : \Delta^{n+1} \to \Delta^n

we dually have functions

d kHom Top(δ k,X):(SingX) n(SingX) n1 d_k \coloneqq Hom_{Top}(\delta_k, X) : (Sing X)_n \to (Sing X)_{n-1}

that send each singular nn-simplex to its kk-face and functions

s kHom Top(σ k,X):(SingX) n(SingX) n+1 s_k \coloneqq Hom_{Top}(\sigma_k,X) : (Sing X)_{n} \to (Sing X)_{n+1}

that regard an nn-simplex as beign a degenerate (“thin”) (n+1)(n+1)-simplex. All these sets of simplices and face and degeneracy maps between them form the following structure.

Definition

A simplicial set SsSetS \in sSet is

  • for each nn \in \mathbb{N} a set S nSetS_n \in Set – the set of nn-simplices;

  • for each injective map δ i:n1¯n¯\delta_i : \overline{n-1} \to \overline{n} of totally ordered sets n¯{0<1<<n}\bar n \coloneqq \{ 0 \lt 1 \lt \cdots \lt n \}

    a function d i:S nS n1d_i : S_{n} \to S_{n-1} – the iith face map on nn-simplices;

  • for each surjective map σ i:n+1¯n¯\sigma_i : \overline{n+1} \to \bar n of totally ordered sets

    a function σ i:S nS n+1\sigma_i : S_{n} \to S_{n+1} – the iith degeneracy map on nn-simplices;

such that these functions satisfy the simplicial identities.

Definition

The simplicial identities satisfied by face and degeneracy maps as above are (whenever these maps are composable as indicated):

  1. d id j=d j1d i d_i \circ d_j = d_{j-1} \circ d_i if i<ji \lt j,

  2. s is j=s js i1s_i \circ s_j = s_j \circ s_{i-1} if i>ji \gt j.

  3. d is j={s j1d i ifi<j id ifi=jori=j+1 s jd i1 ifi>j+1d_i \circ s_j = \left\{ \array{ s_{j-1} \circ d_i & if \; i \lt j \\ id & if \; i = j \; or \; i = j+1 \\ s_j \circ d_{i-1} & if i \gt j+1 } \right.

It is straightforward to check by explicit inspection that the evident injection and restriction maps between the sets of singular simplices make (SingX) (Sing X)_\bullet into a simplicial set. However for working with this, it is good to streamline a little:

Definition

The simplex category Δ\Delta is the full subcategory of Cat on the free categories of the form

[0] {0} [1] {01} [2] {012} . \begin{aligned} [0] & \coloneqq \{0\} \\ [1] & \coloneqq \{0 \to 1\} \\ [2] & \coloneqq \{0 \to 1 \to 2\} \\ \vdots \end{aligned} \,.
Remark

This is called the “simplex category” because we are to think of the object [n][n] as being the “spine” of the nn-simplex. For instance for n=2n = 2 we think of 0120 \to 1 \to 2 as the “spine” of the triangle. This becomes clear if we don’t just draw the morphisms that generate the category [n][n], but draw also all their composites. For instance for n=2n = 2 we have_

[2]={ 1 0 2}. [2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\} \,.
Proposition

A functor

S:Δ opSet S : \Delta^{op} \to Set

from the opposite category of the simplex category to the category Set of sets is canonically identified with a simplicial set, def. 5.

Proof

One checks by inspection that the simplicial identities characterize precisely the behaviour of the morphisms in Δ op([n],[n+1])\Delta^{op}([n],[n+1]) and Δ op([n],[n1])\Delta^{op}([n],[n-1]).

This makes the following evident:

Example

The topological simplices from def. 1 arrange into a cosimplicial object in Top, namely a functor

Δ :ΔTop. \Delta^\bullet : \Delta \to Top \,.

With this now the structure of a simplicial set on (SingX) (Sing X)_\bullet, def. 4, is manifest: it is just the nerve of XX with respect to Δ \Delta^\bullet, namely:

Definition

For XX a topological space its simplicial set of singular simplicies (often called the singular simplicial complex)

(SingX) :Δ opSet (Sing X)_\bullet : \Delta^{op} \to Set

is given by composition of the functor from example 3 with the hom functor of Top:

(SingX):[n]Hom Top(Δ n,X). (Sing X) : [n] \mapsto Hom_{Top}( \Delta^n , X ) \,.
Remark

It turns out – this is the content of the homotopy hypothesis-theorem (Quillen 67) – that homotopy type of the topological space XX is entirely captured by its singular simplicial complex SingXSing X. Moreover, the geometric realization of SingXSing X is a model for the same homotopy type as that of XX, but with the special property that it is canonically a cell complex – a CW-complex. Better yet, SingXSing X is itself already good cell complex, namely a Kan complex. We come to this below.

Simplicial homotopy

The concept of homotopy of morphisms between simplicial sets proceeds in direct analogy with that in topological spaces.

Definition

For XX a simplicial set, def. 5, its simplicial cylinder object is the Cartesian product X×Δ[1]X\times \Delta[1] (formed in the category sSet).

A left homotopy

η:fg \eta \;\colon\; f \Rightarrow g

between two morphisms

f,g:XY f,g\;\colon\; X \longrightarrow Y

of simplicial sets is a morphism

η:X×Δ[1]Y \eta \;\colon\; X \times \Delta[1] \longrightarrow Y

such that the following diagram commutes

X (id X,d 1) f X×Δ 1 η Y (id x,d 0) g X. \array{ X \\ {}^{\mathllap{(id_X,d_1)}}\downarrow & \searrow^{\mathllap{f}} \\ X \times \Delta^1 &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id_x, d_0)}}\uparrow & \nearrow_{\mathllap{g}} \\ X } \,.

For YY a Kan complex, def. 5, its simplicial path space object is the function complex X Δ[1]X^{\Delta[1]} (formed in the category sSet).

A right homotopy

η:fg \eta \;\colon\; f \Rightarrow g

between two morphisms

f,g:XY f,g\;\colon\; X \longrightarrow Y

of simplicial sets is a morphism

η:XY Δ[1] \eta \colon X \longrightarrow Y^{\Delta[1]}

such that the following diagram commutes

Y f Y d 1 X η Y Δ[1] g Y d 0 Y. \array{ && Y \\ & {}^{\mathllap{f}}\nearrow & \uparrow^{\mathrlap{Y^{d_1}}} \\ X &\stackrel{\eta}{\longrightarrow}& Y^{\Delta[1]} \\ & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{Y^{d_0}}} \\ && Y } \,.
Proposition

For YY a Kan complex, def. 15, and XX any simplicial set, then left homotopy, def. 9, regarded as a relation

(fg)(fg) (f\sim g) \Leftrightarrow (f \stackrel{\exists}{\Rightarrow} g)

on the hom set Hom sSet(X,Y)Hom_{sSet}(X,Y), is an equivalence relation.

Definition

A morphism f:XYf \colon X \longrightarrow Y of simplicial sets is a left/right homotopy equivalence if there exists a morphisms XY:gX \longleftarrow Y \colon g and left/right homotopies (def. 9)

gfid X,fgid Y g \circ f \Rightarrow id_X\,,\;\;\;\; f\circ g \Rightarrow id_Y

The the basic invariants of simplicial sets/Kan complexes in simplicial homotopy theory are their simplicial homotopy groups, to which we turn now.

Given that a Kan complex is a special simplicial set that behaves like a combinatorial model for a topological space, the simplicial homotopy groups of a Kan complex are accordingly the combinatorial analog of the homotopy groups of topological spaces: instead of being maps from topological spheres modulo maps from topological disks, they are maps from the boundary of a simplex modulo those from the simplex itself.

Accordingly, the definition of the discussion of simplicial homotopy groups is essentially literally the same as that of homotopy groups of topological spaces. One technical difference is for instance that the definition of the group structure is slightly more non-immediate for simplicial homotopy groups than for topological homotopy groups (see below).

Definition

For XX a Kan complex, then its 0th simplicial homotopy group (or set of connected components) is the set of equivalence classes of vertices modulo the equivalence relation X 1(d 1,d 0)X 0×X 0X_1 \stackrel{(d_1,d_0)}{\longrightarrow} X_0 \times X_0

π 0(X):X 0/X 1. \pi_0(X) \colon X_0/X_1 \,.

For xX 0x \in X_0 a vertex and for nn \in \mathbb{N}, n1n \geq 1, then the underlying set of the nnth simplicial homotopy group of XX at xx – denoted π n(X,x)\pi_n(X,x) – is, the set of equivalence classes [α][\alpha] of morphisms

α:Δ nX \alpha \colon \Delta^n \to X

from the simplicial nn-simplex Δ n\Delta^n to XX, such that these take the boundary of the simplex to xx, i.e. such that they fit into a commuting diagram in sSet of the form

Δ[n] Δ[0] x Δ[n] α X, \array{ \partial \Delta[n] & \longrightarrow & \Delta[0] \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Delta[n] &\stackrel{\alpha}{\longrightarrow}& X } \,,

where two such maps α,α\alpha, \alpha' are taken to be equivalent is they are related by a simplicial homotopy η\eta

Δ[n] i 0 α Δ[n]×Δ[1] η X i 1 α Δ[n] \array{ \Delta[n] \\ \downarrow^{i_0} & \searrow^{\alpha} \\ \Delta[n] \times \Delta[1] &\stackrel{\eta}{\longrightarrow}& X \\ \uparrow^{i_1} & \nearrow_{\alpha'} \\ \Delta[n] }

that fixes the boundary in that it fits into a commuting diagram in sSet of the form

Δ[n]×Δ[1] Δ[0] x Δ[n]×Δ[1] η X. \array{ \partial \Delta[n] \times \Delta[1] & \longrightarrow & \Delta[0] \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Delta[n] \times \Delta[1] &\stackrel{\eta}{\longrightarrow}& X } \,.

These sets are taken to be equipped with the following group structure.

Definition

For XX a Kan complex, for xX 0x\in X_0, for n1n \geq 1 and for f,g:Δ[n]Xf,g \colon \Delta[n] \to X two representatives of π n(X,x)\pi_n(X,x) as in def. 11, consider the following nn-simplices in X nX_n:

v i{s 0s 0s 0(x) for0in2 f fori=n1 g fori=n+1 v_i \coloneqq \left\{ \array{ s_0 \circ s_0 \circ \cdots \circ s_0 (x) & for \; 0 \leq i \leq n-2 \\ f & for \; i = n-1 \\ g & for \; i = n+1 } \right.

This corresponds to a morphism Λ n+1[n]X\Lambda^{n+1}[n] \to X from a horn of the (n+1)(n+1)-simplex into XX. By the Kan complex property of XX this morphism has an extension θ\theta through the (n+1)(n+1)-simplex Δ[n]\Delta[n]

Λ n+1[n] X θ Δ[n+1] \array{ \Lambda^{n+1}[n] & \longrightarrow & X \\ \downarrow & \nearrow_{\mathrlap{\theta}} \\ \Delta[n+1] }

From the simplicial identities one finds that the boundary of the nn-simplex arising as the nnth boundary piece d nθd_n \theta of θ\theta is constant on xx

d id nθ=d n1d iθ=x d_i d_{n} \theta = d_{n-1} d_i \theta = x

So d nθd_n \theta represents an element in π n(X,x)\pi_n(X,x) and we define a product operation on π n(X,x)\pi_n(X,x) by

[f][g][d nθ]. [f]\cdot [g] \coloneqq [d_n \theta] \,.

(e.g. Goerss-Jardine 99, p. 26)

Remark

All the degenerate nn-simplices v 0in2v_{0 \leq i \leq n-2} in def. 12 are just there so that the gluing of the two nn-cells ff and gg to each other can be regarded as forming the boundary of an (n+1)(n+1)-simplex except for one face. By the Kan extension property that missing face exists, namely d nθd_n \theta. This is a choice of gluing composite of ff with gg.

Lemma

The product on homotopy group elements in def. 12 is well defined, in that it is independent of the choice of representatives ff, gg and of the extension θ\theta.

e.g. (Goerss-Jardine 99, lemma 7.1)

Lemma

The product operation in def. 12 yields a group structure on π n(X,x)\pi_n(X,x), which is abelian for n2n \geq 2.

e.g. (Goerss-Jardine 99, theorem 7.2)

Remark

The first homotopy group, π 1(X,x)\pi_1(X,x), is also called the fundamental group of XX.

Definition

For X,YKanCplxsSetX,Y \in KanCplx \hookrightarrow sSet two Kan complexes, then a morphism

f:XY f \colon X \longrightarrow Y

is called a weak homotopy equivalence if it induces isomorphisms on all simplicial homotopy groups, i.e. if

  1. π 0(f):π 0(X)π 0(Y)\pi_0(f) \colon \pi_0(X) \longrightarrow \pi_0(Y) is a bijection of sets;

  2. π n(f,x):π n(X,x)lonrightarrowπ n(Y,f(x))\pi_n(f,x) \colon \pi_n(X,x) \lonrightarrow \pi_n(Y,f(x)) is an isomorphism of groups for all xX 0x\in X_0 and all nn \in \mathbb{N}; n1n \geq 1.

Kan fibrations

Definition

For each ii, 0in0 \leq i \leq n, the (n,i)(n,i)-horn is the subsimplicial set

Λ i[n]Δ[n] \Lambda^i[n] \hookrightarrow \Delta[n]

of the simplicial nn-simplex, which is the union of all faces except the i thi^{th} one.

This is called an outer horn if k=0k = 0 or k=nk = n. Otherwise it is an inner horn.

(graphics taken from Friedman 08)

Remark

Since sSet is a presheaf category, unions of subobjects make sense and they are calculated objectwise, thus in this case dimensionwise. This way it becomes clear what the structure of a horn as a functor Λ k[n]:Δ opSet\Lambda^k[n]: \Delta^{op} \to Set must therefore be: it takes [m][m] to the collection of ordinal maps f:[m][n]f: [m] \to [n] which do not have the element kk in the image.

Definition

A Kan complex is a simplicial set SS that satisfies the Kan condition,

  • which says that all horns of the simplicial set have fillers/extend to simplices;

  • which means equivalently that the unique homomorphism SptS \to pt from SS to the point (the terminal simplicial set) is a Kan fibration;

  • which means equivalently that for all diagrams of the form

    Λ i[n] S Δ[n] ptΛ i[n] S Δ[n] \array{ \Lambda^i[n] &\to& S \\ \downarrow && \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow && \\ \Delta[n] }

    there exists a diagonal morphism

    Λ i[n] S Δ[n] ptΛ i[n] S Δ[n] \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \\ \Delta[n] }

    completing this to a commuting diagram;

  • which in turn means equivalently that the map from nn-simplices to (n,i)(n,i)-horns is an epimorphism

    [Δ[n],S][Λ i[n],S]. [\Delta[n], S]\, \twoheadrightarrow \,[\Lambda^i[n],S] \,.
Proposition

For XX a topological space, its singular simplicial complex Sing(X)Sing(X), def. 8, is a Kan complex, def. 15.

Proof

The inclusions Λ n Top kΔ Top n{{\Lambda^n}_{Top}}_k \hookrightarrow \Delta^n_{Top} of topological horns into topological simplices are retracts, in that there are continuous maps Δ Top nΛ n Top k\Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k given by “squashing” a topological nn-simplex onto parts of its boundary, such that

(Λ n Top kΔ Top nΛ n Top k)=Id. ({{\Lambda^n}_{Top}}_k \to \Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k) = Id \,.

Therefore the map [Δ n,Π(X)][Λ k n,Π(X)][\Delta^n, \Pi(X)] \to [\Lambda^n_k,\Pi(X)] is an epimorphism, since it is equal to to Top(Δ n,X)Top(Λ k n,X)Top(\Delta^n, X) \to Top(\Lambda^n_k, X) which has a right inverse Top(Λ k n,X)Top(Δ n,X)Top(\Lambda^n_k, X) \to Top(\Delta^n, X).

More generally:

Definition

A morphism ϕ:ST\phi \colon S \longrightarrow T in sSet is called a Kan fibration if it has the right lifting property again all horn inclusions, def. 14, hence if for every commuting diagram of the form

Λ i[n] S ϕ Δ[n] T \array{ \Lambda^i[n] &\longrightarrow& S \\ \downarrow && \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T }

there exists a lift

Λ i[n] S ϕ Δ[n] T. \array{ \Lambda^i[n] &\longrightarrow& S \\ \downarrow &\nearrow& \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T } \,.

This is the simplicial incarnation of the concept of Serre fibrations of topological spaces:

Definition

A continuous function f:XYf \colon X \longrightarrow Y between topological spaces is a Serre fibration if for all CW-complexes CC and for every commuting diagram in Top of the form

C X f C×I Y \array{ C &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ C \times I &\longrightarrow& Y }

there exists a lift

C X f C×I Y. \array{ C &\longrightarrow& X \\ \downarrow &\nearrow& \downarrow^{\mathrlap{f}} \\ C \times I &\longrightarrow& Y } \,.
Proposition

A continuous function f:XYf \colon X \longrightarrow Y is a Serre fibration, def. 17, precisely if Sing(f):Sing(X)Sing(Y)Sing(f) \colon Sing(X) \longrightarrow Sing(Y) (def. 8) is a Kan fibration, def. 16.

The proof uses the basic tool of nerve and realization-adjunction to which we get to below in prop. 7.

Proof

First observe that the left lifting property against all CC×IC \hookrightarrow C \times I for CC a CW-complex is equivalent to left lifting against geometric realization |Λ i[n]||Δ[n]|{\vert \Lambda^i[n]\vert} \hookrightarrow {\vert \Delta[n]\vert} of horn inclusions. Then apply the natural isomorphism Top(||,)sSet(,Sing())Top({\vert-\vert},-) \simeq sSet(-,Sing(-)), given by the adjunction of prop. 7 and example 4, to the lifting diagrams.

Lemma

Let p:XYp \colon X \longrightarrow Y be a Kan fibration, def. 16, and let f 1,f 2:AXf_1,f_2 \colon A \longrightarrow X be two morphisms. If there is a left homotopy (def. 9) f 1f 2f_1 \Rightarrow f_2 between these maps, then there is a fiberwise homotopy equivalence, def. 10, between the pullback fibrations f 1 *Xf 2 *Xf_1^\ast X \simeq f_2^\ast X.

(e.g. Goerss-Jardine 99, chapter I, lemma 10.6)

While simplicial sets have the advantage of being purely combinatorial structures, the singular simplicial complex of any given topological space, def. 8 is in general a huge simplicial set which does not lend itself to detailed inspection. The following is about small models.

Definition

A Kan fibration ϕ:ST\phi \colon S \longrightarrow T, def. 16, is called a minimal Kan fibration if for any two cells in the same fiber with the same boundary if they are homotopic relative their boundary, then they are already equal.

More formally, ϕ\phi is minimal precisely if for every commuting diagram of the form

(Δ[n])×Δ[1] p 1 Δ[n] Δ[n]×Δ[1] h S p 1 ϕ Δ[n] T \array{ (\partial \Delta[n]) \times \Delta[1] &\stackrel{p_1}{\longrightarrow}& \partial \Delta[n] \\ \downarrow && \downarrow \\ \Delta[n] \times \Delta[1] &\stackrel{h}{\longrightarrow}& S \\ \downarrow^{\mathrlap{p_1}} && \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T }

then the two composites

Δ[n]d 1d 0Δ[n]×Δ[1]hS \Delta[n] \stackrel{\overset{d_0}{\longrightarrow}}{\underset{d_1}{\longrightarrow}} \Delta[n] \times \Delta[1] \stackrel{h}{\longrightarrow} S

are equal.

Proposition

The pullback (in sSet) of a minimal Kan fibration, def. 18, along any morphism is again a mimimal Kan fibration.

anodyne extensions

(Goerss-Jardine 99, chapter I, section 4, Joyal-Tierney 05, section 31)

Proposition

For every Kan fibration, def. 16, there exists a fiberwise strong deformation retract to a minimal Kan fibration, def. 18.

(e.g. Goerss-Jardine 99, chapter I, prop. 10.3, Joyal-Tierney 05, theorem 3.3.1, theorem 3.3.3).

Proof idea

Choose representatives by induction, use that in the induction step one needs lifts of anodyne extensions against a Kan fibration, which exist.

Lemma

A morphism between minimal Kan fibrations, def. 18, which is fiberwise a homotopy equivalence, def. 10, is already an isomorphism.

(e.g. Goerss-Jardine 99, chapter I, lemma 10.4)

Proof idea

Show the statement degreewise. In the induction one needs to lift anodyne extensions agains a Kan fibration.

Lemma

Every minimal Kan fibration, def. 18, over a connected base is a simplicial fiber bundle, locally trivial over every simplex of the base.

(e.g. Goerss-Jardine 99, chapter I, corollary 10.8)

Proof

By assumption of the base being connected, the classifying maps for the fibers over any two vertices are connected by a zig-zag of homotopies, hence by lemma 3 the fibers are connected by homotopy equivalences and then by prop. 5 and lemma 4 they are already isomorphic. Write FF for this typical fiber.

Moreover, for all nn the morphisms Δ[n]Δ[0]Δ[n]\Delta[n] \to \Delta[0] \to \Delta[n] are left homotopic to Δ[n]idΔ[n]\Delta[n] \stackrel{id}{\to} \Delta[n] and so applying lemma 3 and prop. 4 once more yields that the fiber over each Δ[n]\Delta[n] is isomorphic to Δ[n]×F\Delta[n]\times F.

Geometric realization

So far we we have considered passing from topological spaces to simplicial sets by applying the singular simplicial complex functor of def. 8. Now we discuss a left adjoint of this functor, called geometric realization, which turns a simplicial set into a topological space by identifying each of its abstract n-simplices with the standard topological nn-simplex.

This is an example of a general abstract phenomenon:

Proposition

Let

δ:D𝒞 \delta \;\colon\; D \longrightarrow \mathcal{C}

be a functor from a small category DD to a locally small category 𝒞\mathcal{C} with all colimits. Then the nerve-functor

N:𝒞[D op,Set] N \;\colon\; \mathcal{C} \longrightarrow [D^{op}, Set]
N(X)𝒞(δ(),X) N(X) \coloneqq \mathcal{C}(\delta(-),X)

has a left adjoint functor ||{\vert-\vert}, called geometric realization,

(||N):𝒞N||[D op,Set] ({\vert-\vert} \dashv N) \;\colon\; \mathcal{C} \stackrel{\overset{{\vert-\vert}}{\longleftarrow}}{\underset{N}{\longrightarrow}} [D^{op}, Set]

given by the coend

|S|= dDδ(d)S(d). {\vert S\vert} = \int^{d \in D} \delta(d) \cdot S(d) \,.

(Kan 58)

Proof

By basic propeties of ends and coends:

[D op,Set](S,N(X)) = dDSet(S(d),N(X)(d)) = dDSet(S(d),𝒞(δ(d),X)) dD𝒞(δ(d)S(d),X) 𝒞( dDδ(d)S(d),X) =𝒞(|S|,X). \begin{aligned} [D^{op}, Set](S,N(X)) & = \int_{d \in D} Set(S(d), N(X)(d)) \\ & = \int_{d\in D} Set(S(d), \mathcal{C}(\delta(d),X)) \\ & \simeq \int_{d \in D} \mathcal{C}(\delta(d) \cdot S(d), X) \\ & \simeq \mathcal{C}(\int^{d \in D} \delta(d) \cdot S(d), X) \\ & = \mathcal{C}({\vert S\vert}, X) \,. \end{aligned}
Example

The singular simplicial complex functor SingSing of def. 8 has a left adjoint geometric realization functor

||:sSetTop {\vert-\vert} \colon sSet \longrightarrow Top

given by the coend

|S|= [n]ΔΔ nS n. {\vert S \vert} = \int^{[n]\in \Delta} \Delta^n \cdot S_n \,.

Topological geometric realization takes values in particularly nice topological spaces.

Proposition

The topological geometric realization of simplicial sets in example 4 takes values in CW-complexes.

(e.g. Goerss-Jardine 99, chapter I, prop. 2.3)

Thus for a topological space XX the adjunction counit ϵ X:|SingX|X\epsilon_X \colon {\vert Sing X\vert} \longrightarrow X of the nerve and realization-adjunction is a candidate for a replacement of XX by a CW-complex. For this, ϵ X\epsilon_X should be at least a weak homotopy equivalence, i.e. induce isomorphisms on all homotopy groups. Since homotopy groups are built from maps into XX out of compact topological spaces it is plausible that this works if the topology of XX is entirely detected by maps out of compact topological spaces into XX. Topological spaces with this property are called compactly generated.

We take compact topological space to imply Hausdorff topological space.

Definition

A subspace UXU \subset X of a topological space XX is called compactly open or compactly closed, respectively, if for every continuous function f:KXf \colon K \longrightarrow X out of a compact topological space the preimage f 1(U)Kf^{-1}(U) \subset K is open or closed, respectively.

A topological space XX is a compactly generated topological space if each of its compactly closed subspaces is already closed.

Write

Top cgTop Top_{cg} \hookrightarrow Top

for the full subcategory of Top on the compactly generated topological spaces.

Often the condition is added that a compactly closed topological space be also a weakly Hausdorff topological space.

Corollary

The topological geometric realization functor of simplicial sets in example 4 takes values in compactly generated topological spaces

||:sSetTop cg {\vert - \vert} \;\colon\; sSet \longrightarrow Top_{cg}
Proof

By example 5 and prop. 19.

Proposition

The subcategory Top cgTopTop_{cg} \hookrightarrow Top of def. 20 has the following properties

  1. It is a coreflective subcategory

    Top cgkTop. Top_{cg} \stackrel{\hookrightarrow}{\underset{k}{\longleftarrow}} Top \,.

    The coreflection k(X)k(X) of a topological space is given by adding to the open subsets of XX all compactly open subsets, def. 20.

  2. It has all small limits and colimits.

    The colimits are computed in TopTop, the limits are the image under kk of the limits as computed in TopTop.

  3. It is a cartesian closed category.

    The cartesian product in Top cgTop_{cg} is the image under kk of the Cartesian product formed in TopTop.

This is due to (Steenrod 67), expanded on in (Lewis 78, appendix A). One says that prop. 8 with example 5 makes Top cgTop_{cg} a “convenient category of topological spaces”.

Proposition

Regarded, via corollary 1 as a functor ||:sSetTop cg{\vert - \vert} \colon sSet \to Top_{cg}, geometric realization preserves finite limits.

See at Geometric realization is left exact.

Proof idea

The key step in the proof is to use the cartesian closure of Top cgTop_{cg} (prop. 8). This gives that the Cartesian product is a left adjoint and hence preserves colimits in each variable, so that the coend in the definition of the geometric realization may be taken out of Cartesian products.

Lemma

The geometric realization, example 4, of a minimal Kan fibration, def. 18 is a Serre fibration, def. 17.

This is due to (Gabriel-Zisman 67). See for instance (Goerss-Jardine 99, chapter I, corollary 10.8, theorem 10.9).

Proof idea

By prop. 5 minimal Kan fibrations are simplicial fiber bundles, locally trivial over each simplex in the base. By prop. 9 this property translates to their geometric realization also being a locally trivial fiber bundle of topological spaces, hence in particular a Serre fibration.

Proposition

The geometric realization, example 4, of any Kan fibration, def. 16 is a Serre fibration, def. 17.

This is due to (Quillen 68). See for instance (Goerss-Jardine 99, chapter I, theorem 10.10).

Proposition

For SS a Kan complex, then the unit of the nerve and realization-adjunction (prop. 7, example 4)

SSing|S| S \longrightarrow Sing {\vert S \vert}

is a weak homotopy equivalence, def. 13.

For XX any topological space, then the adjunction counit

|SingX|X {\vert Sing X\vert} \longrightarrow X

is a weak homotopy equivalence

e.g. (Goerss-Jardine 99, chapter I, prop. 11.1 and p. 63).

Proof idea

Use prop. 4 and prop. 10 applied to the path fibration to proceed by induction.

The classical model structure sSet QuillensSet_{Quillen}

Definition

The classical model structure on simplicial sets, sSet QuillensSet_{Quillen}, has the following distinguished classes of morphisms:

Quillen equivalence with Top QuillenTop_{Quillen}

Theorem

The singular simplicial complex/geometric realization-adjunction of example 4 constitutes a Quillen equivalence of the classical model structure sSet QuillensSet_{Quillen} of def. 21 with the classical model structure on topological spaces:

(||Sing):Top QuillenSing||sSet Quillen ({\vert -\vert}\dashv Sing) : Top_{Quillen} \stackrel{\overset{{\vert -\vert}}{\leftarrow}}{\underset{Sing}{\to}} sSet_{Quillen}
Proof

First of all, the adjunction is indeed a Quillen adjunction: prop. 4 says in particular that Sing()Sing(-) takes Serre fibrations to Kan fibrations and prop. 19 gives that ||{\vert-\vert} sends monomorphisms of simplicial sets to relative cell complexes.

Now prop. 11 says that the derived adjunction unit and counit are weak equivalences, and hence the Quillen adjunction is a Quillen equivalence.

Properties

Basic properties

Proposition

In model structure sSet QuillensSet_{Quillen}, def. 21, the following holds.

Proposition

The acyclic fibrations in sSet QuillensSet_{Quillen}(i.e. the maps that are both fibrations as well as weak equivalences) between Kan complexes are precisely the morphisms f:XYf : X \to Y that have the right lifting property with respect to all inclusions Δ[n]Δ[n]\partial \Delta[n] \hookrightarrow \Delta[n] of boundaries of nn-simplices into their nn-simplices

Δ[n] X f Δ[n] Y. \array{ \partial \Delta[n] &\to& X \\ \downarrow &{}^\exists\nearrow& \downarrow^f \\ \Delta[n] &\to& Y } \,.

This appears spelled out for instance as (Goerss-Jardine 99, theorem 11.2).

In fact:

Proposition

sSet QuillensSet_{Quillen} is a cofibrantly generated model category with

  • generating cofibrations the boundary inclusions Δ[n]Δ[n]\partial \Delta[n] \to \Delta[n];

  • generating acyclic cofibrations the horn inclusions Λ i[n]Δ[n]\Lambda^i[n] \to \Delta[n].

Theorem

Let WW be the smallest class of morphisms in sSetsSet satisfying the following conditions:

  1. The class of monomorphisms that are in WW is closed under pushout, transfinite composition, and retracts.
  2. WW has the two-out-of-three property in sSetsSet and contains all the isomorphisms.
  3. For all natural numbers nn, the unique morphism Δ[n]Δ[0]\Delta [n] \to \Delta [0] is in WW.

Then WW is the class of weak homotopy equivalences.

Proof
  • First, notice that the horn inclusions Λ 0[1]Δ[1]\Lambda^0 [1] \hookrightarrow \Delta [1] and Λ 1[1]Δ[1]\Lambda^1 [1] \hookrightarrow \Delta [1] are in WW.
  • Suppose that the horn inclusion Λ k[m]Δ[m]\Lambda^k [m] \hookrightarrow \Delta [m] is in WW for all m<nm \lt n and all 0km0 \le k \le m. Then for 0ln0 \le l \le n, the horn inclusion Λ l[n]Δ[n]\Lambda^l [n] \hookrightarrow \Delta [n] is also in WW.
  • Quillen’s small object argument then implies all the trivial cofibrations are in WW.
  • If p:XYp : X \to Y is a trivial Kan fibration, then its right lifting property implies there is a morphism s:YXs : Y \to X such that ps=id Yp \circ s = id_Y, and the two-out-of-three property implies s:YXs : Y \to X is a trivial cofibration. Thus every trivial Kan fibration is also in WW.
  • Every weak homotopy equivalence factors as pip \circ i where pp is a trivial Kan fibration and ii is a trivial cofibration, so every weak homotopy equivalence is indeed in WW.
  • Finally, noting that the class of weak homotopy equivalences satisfies the conditions in the theorem, we deduce that it is the smallest such class.

As a corollary, we deduce that the classical model structure on sSetsSet is the smallest (in terms of weak equivalences) model structure for which the cofibrations are the monomorphisms and the weak equivalences include the (combinatorial) homotopy equivalences.

Proposition

Let π 0:sSetSet\pi_0 : sSet \to Set be the connected components functor, i.e. the left adjoint of the constant functor cst:SetsSetcst : Set \to sSet. A morphism f:ZWf : Z \to W in sSetsSet is a weak homotopy equivalence if and only if the induced map

π 0K f:π 0K Wπ 0K Z\pi_0 K^f : \pi_0 K^W \to \pi_0 K^Z

is a bijection for all Kan complexes KK.

Proof

One direction is easy: if KK is a Kan complex, then axiom SM7 for simplicial model categories implies the functor K ():sSet opsSetK^{(-)} : sSet^{op} \to sSet is a right Quillen functor, so Ken Brown’s lemma implies it preserves all weak homotopy equivalences; in particular, π 0K ():sSet opSet\pi_0 K^{(-)} : sSet^{op} \to Set sends weak homotopy equivalences to bijections.

Conversely, when KK is a Kan complex, there is a natural bijection between π 0K X\pi_0 K^X and the hom-set Ho(sSet)(X,K)Ho (sSet) (X, K), and thus by the Yoneda lemma, a morphism f:ZWf : Z \to W such that the induced morphism π 0K Wπ 0K Z\pi_0 K^W \to \pi_0 K^Z is a bijection for all Kan complexes KK is precisely a morphism that becomes an isomorphism in Ho(sSet)Ho (sSet), i.e. a weak homotopy equivalence.

Properness

The Quillen model structure is both left and right proper. Left properness is automatic since all objects are cofibrant. Right properness follows from the following argument: it suffices to show that there is a functor RR which (1) preserves fibrations, (2) preserves pullbacks of fibrations, (3) preserves and reflects weak equivalences, and (4) lands in a category in which the pullback of a weak equivalence along a fibration is a weak equivalence. For if so, we can apply RR to the pullback of a fibration along a weak equivalence to get another such pullback in the codomain of RR, which is a weak equivalence, and hence the original pullback was also a weak equivalence. Two such functors RR are

This may be found, for instance, in II.8.6–7 of Goerss-Jardine. Another proof may be found in Moss, and a different proof of properness may be found in Cisinski, Prop. 2.1.5.

References

The original article is

  • Dan Quillen, chapter II, section 3 of Homotopical algebra, Lecture Notes in Mathematics 43, Springer-Verlag 1967, iv+156 pp.

The proof there is purely combinatorial (i.e. does not use topological spaces): he uses the theory of minimal Kan fibrations, the fact that the latter are fiber bundles, as well as the fact that the classifying space of a simplicial group is a Kan complex. This proof has been rewritten several times in the literature: at the end of

A proof (in fact two variants of it) using the Kan fibrant replacement Ex Ex^\infty functor is given

which discusses the topic as a special case of a Cisinski model structure.

The fun part is not that much about the existence of model structure, but to prove that the fibrations are precisely the Kan fibrations (and also to prove all the good properties of Ex Ex^\infty without using topological spaces); for two different proofs of this fact using Ex Ex^\infty, see Prop. 2.1.41 as well as Scholium 2.3.21 for an alternative). For the rest, everything was already in the book of Gabriel and Zisman, for instance.

Another approach also using Ex Ex^\infty is in

  • Sean Moss, Another approach to the Kan-Quillen model structure, arXiv.

A standard textbook references for the classical model structure is

A proof of the model structure no relying on the classical model structure on topological spaces nor on explicit models for Kan fibrant replacement is givn in

Revised on November 5, 2017 01:50:36 by Mike Shulman (76.167.222.204)