homotopy theory

# Contents

## Idea

Recall that a simplicial set is a combinatorial model for a topological space. This relation is most immediate when the simplicial set is in fact a Kan complex (an ∞-groupoid).

A simplicial group is a simplicial set with the structure of a group on it. It turns out that this necessarily means that it is also a Kan complex. Therefore a simplicial group is

• an ∞-groupoid with an extra group structure on it;

• a model for a topological space with a group structure.

Accordingly (as discussed at group) a simplicial group $G$ gives rise to

• a one-object $\infty$-groupoid $\mathbf{B} G$ whose explicit standard realization as a simplicial set is denoted $\bar W G$

• an $\infty$-groupoid $\mathbf{E} G$ whose explicit standard realization as a simplicial set (even a simplicial group, again) is denoted $W G$

• such that there is a fibration

$\array{ \mathbf{E} G &:=& W G \\ \downarrow && \downarrow \\ \mathbf{B} G &:=& \bar W G }$

which is the universal G-bundle.

Simplicial abelian groups are models for connective modules over the Eilenberg-Mac Lane spectrum $H \mathbf{Z}$; see Dold-Kan correspondence and stable Dold-Kan correspondence.

## Definition

A simplicial group, $G$, is a simplicial object in the category Grp of groups.

The category of simplicial groups is the category of functors from $\Delta^{op}$ to Grp. It will be denoted $\Simp\Grp$.

## Properties

### As Kan complexes

###### Theorem (J. C. Moore)

The simplicial set underlying any simplicial group (by forgetting the group structure) is a Kan complex.

This is due to (Moore, 1954)

In fact, not only are the horn fillers guaranteed to exist, but there is an algorithm that provides explicit fillers. This implies that constructions on a simplicial group that use fillers of horns can often be adjusted to be functorial by using the algorithmically defined fillers. An argument that just uses ‘existence’ of fillers can be refined to give something more ‘coherent’.

###### Proof

Let $G$ be a simplicial group.

Here is the explicit algorithm that computes the horn fillers:

Let $(y_0,\ldots, y_{k-1}, -,y_{k+1}, \ldots, y_n)$ give a horn in $G_{n-1}$, so the $y_i$s are $(n-1)$ simplices that fit together as if they were all but one, the $k^{th}$ one, of the faces of an $n$-simplex. There are three cases:

1. if $k = 0$:

• Let $w_n = s_{n-1}y_n$ and then $w_i = w_{i+1}(s_{i-1}d_i w_{i+1})^{-1}s_{i-1}y_i$ for $i = n, \ldots, 1$, then $w_1$ satisfies $d_i w_1 = y_i$, $i\neq 0$;
2. if $0\lt k \lt n$:

• Let $w_0 = s_0 y_0$ and $w_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i$ for $i = 0, \ldots, k-1$, then take $w_n = w_{k-1}(s_{n-1}d_nw_{k-1})^{-1}s_{n-1}y_n$, and finally a downwards induction given by $w_i = w_{i+1}(s_{i-1}d_{i}w_{i+1})^{-1}s_{i-1}y_i$, for $i = n, \ldots, k+1$, then $w_{k+1}$ gives $d_{i}w_{k+1} = y_i$ for $i \neq k$;
3. if $k=n$:

• use $w_0 = s_0 y_0$ and $w_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i$ for $i = 0, \ldots, n-1$, then $w_{n-1}$ satisfies $d_i w_{n-1} = y_i$, $i\neq n$.
###### Remarks
• The filler for any horn can be chosen to be a product of degenerate elements.

• The simplicial homotopy groups of a simplicial group, $G$, can be calculated as the homology groups of the Moore complex of $G$. This is, in general, a non-Abelian chain complex.

• A simplicial group can be considered as a simplicial groupoid having exactly one object. If $G$ is a simplicial group, the suggested notation for the corresponding simplicially enriched groupoid would be $\mathbf{B}G$ according to notational conventions suggested elsewhere in the nLab.

• There is a functor due to Dwyer and Kan, called the Dwyer-Kan loop groupoid that takes a simplicial set to a simplicial groupoid. This has a left adjoint $\overline{W}$ (see below), called the classifying space functor, and together they give an equivalence of categories between the homotopy category of simplical sets and that of simplicial groupoids. We thus have that all homotopy types are modelled by simplicial groupoids … and for connected homotopy types by simplicial groups. One important fact to note in this equivalence is that it shifts dimension by 1, so if $G(K)$ is the simplicial group corresponding to the connected simplicial set $K$ then $\pi_k(K)$ is the same as $\pi_{k-1}(G(K))$. This is important when considering algebraic models for a homotopy n-type.

### Fiber sequences

###### Proposition

Let $G$ be a simplicial group and $G_0$ its group of 0-cells, regarded as a simplicially constant simplicial group. Write $G/G_0$ for the evident quotient of simplicial groups.

The evident morphisms

$G_0 \to G \to G/G_0 \simeq \mathbf{B} \Omega G/G_0$

form a fiber sequence in sSet.

###### Proof

One checks that for $X$ any simplicial set and $G$ a simplicial group acting freely on it, the quotient map

$X \to X/G$

is a Kan fibration. This is for instance (Weibel, exercise 8.2.6). By the disucssion at fiber sequence it is therefore sufficient to observe that

$\array{ G_0 &\to& * \\ \downarrow && \downarrow \\ G &\to& G/G_0 }$

is an ordinary pullback of simplicial sets. This is clear since the action of $G_0$ on $G$ is degreewise free (being the action of a subgroup).

###### Example

Let $(G_1 \stackrel{\delta}{\to} G_0)$ be a crossed module of groups, write

$[G_1 \stackrel{\delta}{\to} G_0] = \left( G_0 \times G_1 \stackrel{\overset{(\delta p_2)\cdot p_1}{\to}}{\underset{p_1}{\to}} G_0 \right)$

for groupoid which is the corresponding strict 2-group and write $N[G_1 \to G_0]$ for the nerve being the corresponding simplicial group. Then the above says that

$G_0 \to [G_1 \stackrel{\delta}{\to} G_0] \to \mathbf{B}G_1$

is a fiber sequence of groupoids.

### Free simplicial groups

###### Lemma
$U : AbSGrpg \to SSet$

from simplicial abelian groups to the underlying simplicial sets has a left adjoint

$\mathbb{Z} : SSet \to AbSimpGrp$

from simplicial sets to abelian simplicial groups, the free simplicial abelian group functor that sends the set $X_n$ of $n$-simplices to the free abelian group $(\mathbb{Z}X)_n = \mathbb{Z} X_n$ over it.

This functor $\mathbb{Z}$ has the following properties:

• it preserves weak equivalences

• $\mathbb{Z} X$ is a cofibrant simplicial group

(…)

### Looping and delooping

Let $sSet_0 \hookrightarrow$ sSet be the category of reduced simplicial sets (simplicial sets with a single 0-cell).

###### Definition

For $X \in sSet_0$ define $\Omega X \in sGrpd$ by

$\Omega X : [n] \mapsto (F X_{n+1})/ s_0 F(X_n)$

and

$\Omega X : ([n] \to [k]) \mapsto ...$

### As $\infty$-groups

Simplicial groups are models for ∞-groups. This is exhibited by the model structure on simplicial groups. See also models for group objects in ∞Grpd.

Another equivalent model is that of connected Kan complexes.

At the abstract level of (∞,1)-category theory this equivalence is induced by forming loop space objects and delooping

$\Omega : \infty Grpd_{con} \stackrel{\leftarrow}{\to} \infty Grp : \mathbf{B} \,.$

This (∞,1)-equivalence is modeled by a Quillen equivalence of model categories whose right adjoint Quillen functor is the operation $\overline{W}$ discussed above.

$\mathcal{G} : sSet_0 \stackrel{\stackrel{\simeq_{Quillen}}{\longleftarrow}}{\longrightarrow} sGrp : \overline{W} \,.$

This is for instance in GoerssJardine, chapter 5.

### Closed monoidal structure

The category $sAb$ of simplicial abelian groups is naturally a monoidal category, with the tensor product being degreewise that of abelian groups. This is indeed a closed monoidal category. For $A, B$ The internal hom $[A,B]$ is the simplicial abelian group whose underlying simplicial set is

$[A,B] : [n] \mapsto Hom_{sAb}(A \otimes \mathbb{Z}[\Delta[n]], B) \,,$

where $\mathbf{Z}[-] : sSet \to sAb$ is degreewise the free abelian group functor.

## Delooping and simplicial principal bundles

For $G$ a simplicial group, we describe its delooping Kan complex $\mathbf{B}G \in sSet$ and the corresponding generalized universal bundle $\mathbf{E}G \to \mathbf{B}G$ such that the ordinary pullback

$\array{ P_\bullet &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ X_\bullet &\stackrel{g}{\to}& \mathbf{B}G }$

in sSet models the homotopy pullback in $sSet_{Quillen}$ / (∞,1)-pullback in ∞Grpd

$\array{ P_\bullet &\to& * \\ \downarrow &\swArrow& \downarrow \\ X_\bullet &\to& \mathbf{B}G } \,.$

in the standard model structure on simplicial sets and hence produces the principal ∞-bundle $P_\bullet \to X_\bullet$ classified by $X_\bullet \to \mathbf{B}G$.

For all these constructions exist very explicit combinatorial formulas that go by the symbols

• $\overline{W}G$ for the delooping $\mathbf{B}G$

• $W G$ for the generalized universal bundle $\mathbf{E}G$

• $\tau : X_\bullet \to G_\bullet$ (called the twisting function) for the cocycle $X_\bullet \to \mathbf{B}G$;

• $X_\bullet \times_g W G$ for $P_\bullet$ (called twisted Cartesian product ).

All of these constructions are functorial and hence lift from the context of simplicial sets to that of simplicial presheaves over some site $C$. There they provide models for strict group objects, delooping and principal ∞-bundles in the corresponding (∞,1)-toposes over $C$. In particular in the projective model structure on $[C^{op}, sSet]$ the pullback of the objectwise $W G \to \overline{W}G$ is still a homotopy pullback and models the corresponding principal $\infty$-bundles.

### Delooping

A simplicial group $G$ is a group object internal to the category of Kan complexes. Accordingly, there should be a Kan complex $\mathbf{B}G$ which is the delooping of $G$, i.e. a Kan complex with an essentially unique object, such that the loop space object of that Kan complex reproduces $G$.

An explicit construction of $\mathbf{B}G$ from $G$ goes traditionally by the symbol $\bar W G \in KanCplx$. Another one by $d B G$.

#### Delooping modeled by $\bar W G$

It is immediate to deloop the simplicial group $G$ to the simplicial groupoid that in degree $k$ is the 1-groupoid with a single object and $G_k$ as its collection of morphisms.

###### Definition

For $\mathcal{G}$ a simplicial groupoid that on objects is a constant simplicial set, define a simplicial set $\bar W \mathcal{G}$ as follows.

• $(\overline{W}\mathcal{G})_0 := ob(\mathcal{G}_0)$, the set of objects of the groupoid of 0-simplices (and hence of the groupoid at each level);

• $(\overline{W}\mathcal{G})_1 = Mor(\mathcal{G}_0)$, the collection of morphisms of the groupoid $\mathcal{G}_0$:

and for $n \geq 2$,

• $(\overline{W}\mathcal{G})_n = \{(h_{n-1}, \ldots ,h_0)| h_i \in Mor(\mathcal{G}_i)$ and $s(h_{i-1}) = t(h_i), 0\lt i\lt n\}$.

Here $s$ and $t$ are generic symbols for the domain and codomain mappings of all the groupoids involved. The face and degeneracy mappings between $\overline{W}(\mathcal{G})_1$ and $\overline{W}(\mathcal{G})_0$ are the source and target maps and the identity maps of $\mathcal{G}_0$, respectively; whilst the face and degeneracy maps at higher levels are given as follows:

The face and degeneracy maps are given by

• $d_0(h_{n-1}, \ldots, h_0) = (h_{n-2}, \ldots, h_0)$;

• for $0 \lt i\lt n$, $d_i(h_{n-1}, \ldots, h_0) = (d_{i-1}h_{n-1}, d_{i-2}h_{n-2}, \ldots, d_0h_{n-i}h_{n-i-1},h_{n-i-2}, \ldots , h_0)$;

and

• $d_n(h_{n-1}, \ldots, h_0) = (d_{n-1}h_{n-1}, d_{n-2}h_{n-2}, \ldots, d_1h_{1})$;

whilst

• $s_0(h_{n-1}, \ldots, h_0) = (id_{dom(h_{n-1})},h_{n-1}, \ldots, h_0)$;

and,

• for $0\lt i \leq n$, $s_i(h_{n-1}, \ldots, h_0) = (s_{i-1}h_{n-1}, \ldots, s_0h_{n-i}, id_{cod(h_{n-i})},h_{n-i-1}, \ldots, h_0)$.
###### Definition

For $G$ a simplicial group and $\mathcal{G}$ the corresponding one-object simplicial groupoid, one writes

$\overline{W}G := \overline{W}\mathcal{G} \,.$
###### Remark

The above construction has a straightforward internalization to contexts other than Set. For instance if $G$ is a simplicial object in topological groups or in Lie groups, then $\overline{W}G$ with

$(\overline{W}G)_n := G_{n-1} \times G_{n-2} \times \cdots \times G_0$

is a simplicial object in this context (topological spaces, smooth manifolds, etc.)

In particular, if $C$ is a small category and $G : C^{op} \to sSet$ is a simplicial presheaf that is objectwise a simplicial group, then we have the simplicial presheaf

$\overline{W}G : c \mapsto \overline{W}(G(c)) \,.$

#### Delooping modeled by $d B G$

For $G$ a simplicial group, write $B G$ for the bisimplicial set obtained by taking degreewise the nerve of the delooping groupoid. Write $d B G \in$ sSet for its delooping.

###### Theorem

There is a weak homotopy equivalence

$\bar W G \simeq d B G \,.$

This is shown for instance in (JardineLuo) and in (CegarraRemedios).

#### Examples

If $G$ is an ordinary group, regarded as a simplicially constant simplicial group, then $\overline{W}G$ is the usual bar complex of $G$:

$\overline{W}G = \left( \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} * \right) \,.$

### Cocycles

For $X_\bullet$ a simplicial set a morphism

$g : X_\bullet \to \overline{W}G$

in sSet corresponds precisely to what is called a twisting function, a family of maps

$\{\phi(g)_n : X_n \to G_{n-1}\}$

satisfying the relations

$\array{ d_0 \phi(x) = \phi(d_1 x)(\phi(d_0 x))^{-1} \\ d_i \phi(x) = \phi(d_{i+1}x), i\gt 0, \\ s_i\phi(x) = \phi(s_{i+1}x), i\geq 0, \\ \phi(s_0 x) = 1_G. }$

### Simplicial Principal bundles

Simplicial groups model all ∞-groups in ∞Grpd. Accordingly all principal ∞-bundles in ∞Grpd should be modeled by simplicial principal bundles.

###### Definition

(principal action)

Let $G$ be a simplicial group. For $P$ a Kan complex, an action of $G$ on $E$

$\rho : E \times G \to E$

is called principal if it is degreewise principal, i.e. if for all $n \in \mathbb{N}$ the only elements $g \in G_n$ that have any fixed point $e \in E_n$ in that $\rho(e,g) = e$ are the neutral elements.

###### Example

The canonical action

$G \times G \to G$

of any simplicial group on itself is principal.

###### Definition

(simplicial principal bundle)

For $G$ a simplicial group, a morphism $P \to X$ of Kan complexes equipped with a $G$-action on $P$ is called a $G$-simplicial principal bundle if

• the action is principal;

• the base is isomorphic to the quotient $E/G := \lim_{\to}(E \times G \stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to} E})$ by the action:

$E/G \simeq X \,.$
###### Proposition

A simplicial $G$-principal bundle $P \to X$ is necessarly a Kan fibration.

###### Proof

This appears as Lemma 18.2 in MaySimpOb.

#### Universal simplicial $G$-principal bundle

###### Definition

For $G$ a simplicial group, define the simplicial set $W G$ to be the decalage of $\overline{W}G$

$W G := Dec \overline{W}G \,.$

By the discussion at homotopy pullback this means that for $X_\bullet$ any Kan complex, an ordinary pullback diagram

$\array{ P_\bullet &\to& W G \\ \downarrow && \downarrow \\ X_\bullet &\stackrel{g}{\to}& \overline{W}G }$

in sSet exhibits $P_\bullet$ as the homotopy pullback in $sSet_{Quillen}$ / (∞,1)-pullback in ∞Grpd

$\array{ P_\bullet &\to& * \\ \downarrow &\swArrow& \downarrow \\ X_\bullet &\stackrel{g}{\to}& \overline{W}G } \,,$

i.e. as the homotopy fiber of the cocycle $g$.

###### Definition

We call $P_\bullet := X_\bullet \times^g W G$ the simplicial $G$-principal bundle corresponding to $g$.

###### Proposition

Let $\{\phi : X_n \to G_{(n-1)}\}$ be the twisting function corresponding to $g : X_\bullet \to \overline{W}G$ by the above discussion.

Then the simplicial set $P_\bullet := X_\bullet \times_{g} W G$ is explicitly given by the formula called the twisted Cartesian product $X_\bullet \times^\phi G_\bullet$:

its cells are

$P_n = X_n \times G_n$

with face and degeneracy maps

• $d_i (x,g) = (d_i x , d_i g)$ if $i \gt 0$

• $d_0 (x,g) = (d_0 x, \phi(x) d_0 g)$

• $s_i (x,g) = (s_i x, s_i g)$.

#### References

Here are some pointers on where precisely in the literature the above statements can be found.

One useful reference is

There the abbreviation PCTP ( principal twisted cartesian product ) is used for what above we called just twisted Cartesian products.

The fact that every PTCP $X \times_\phi G \to X$ defined by a twisting function $\phi$ arises as the pullback of $W G \to \overline{W}G$ along a morphjism of simplicial sets $X \to \overline{W}G$ can be found there by combining

1. the last sentence on p. 81 which asserts that pullbacks of PTCPs $X \times_\phi G \to X$ along morphisms of simplicial sets $f : Y \to X$ yield PTCPs corresponding to the composite of $f$ with $\phi$;

2. section 21 which establishes that $W G \to \bar W G$ is the PTCP for some universal twisting function $r(G)$.

3. lemma 21.9 states in the language of composites of twisting functions that every PTCP comes from composing a cocycle $Y \to \bar W G$ with the universal twisting function $r(G)$. In view of the relation to pullbacks in item 1, this yields the statement in the form we stated it above.

An explicit version of the statement that twisted Cartesian products are nothing but pullbacks of a generalized universal bundle is on page 148 of

On page 239 there it is mentioned that

$G \to W G \to \overline{W}G$

is a model for the loop space object fiber sequence

$G \to * \to \mathbf{B}G \,.$

One place in the literature where the observation that $W G$ is the decalage of $\overline{W}G$ is mentioned fairly explicitly is page 85 of

• John Duskin, Simplicial methods and the interpretation of “triple” cohomology, number 163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc. (1975)

## References

A standard reference for the case of abelian simplicial groups is chapter 5 of

Also chapter IV of

and chapter 8 of

• Charles Weibel, An introduction to homological algebra Cambridge (1994)

The algorithm for finding the horn fillers in a simplicial group is given in the proof of theorem 17.1, page 67 there.

This proof that simplicial groups are Kan complexes is originally due to theorem 3.4 in

• John Moore, Semi-Simplicial Complexes And Postnikov Systems, inSymposium International De Topologia Algebraica_ , 1956 conference, book published in 1958

which apears in more detail as theorem 3 on p. 18-04 of

• John Moore, Homotopie des complexes monoideaux, I , Seminaire Henri Cartan, 1954-55. (numdam)

and is often attributed to

• John Moore, Algebraic homotopy theory, lecture notes, Princeton University, 1955–1956

In fact, it seems that this is the origin of the very notion of Kan complex.

A proof is also on p. 14 of

Section 1.3.3 of

discusses simplicial groups in the context of nonabelian algebraic topology.

Revised on November 22, 2016 09:36:22 by Urs Schreiber (195.37.209.183)