Contents

group theory

# Contents

## Idea

For $G$ a simplicial group, there is a reduced simplicial set, traditionally denoted $\overline W G$ and called the classifying space or classifying complex of $G$, which is a model for the delooping of $G$ and such that the functor $\overline{W}(-)$ is right adjoint to the standard simplicial loop space-construction $G$ (here denoted by $L$ to avoid a clash of notations).

$SimplicialGroups \underoverset {\;\;\;\underset{\overline{W}}{\longrightarrow}\;\;\;} {\;\;\;\overset{L}{\longleftarrow}\;\;\;} {\bot} SimplicialSets_{red}$

is a Quillen equivalence (Prop. below) between the model structure on simplicial groups and the model structure on reduced simplicial sets, modelling looping and delooping of homotopy types in simplicial homotopy theory.

## Definition

### In components

###### Definition

(standard universal principal simplicial complex)
For $G$ a simplicial group, one writes

$W G \;\in\; SimplicialSets$

for the the simplicial set whose

• underlying sets are

$(W G)_n \;\coloneqq\; G_{n} \times G_{n-1} \times \cdots \times G_0$
• face maps are given by

(1)\begin{aligned} & d_i \big( g_n, g_{n-1}, \cdots, g_0 \big) \\ & \;\coloneqq\; \left\{ \array{ \big( d_i(g_n), \, d_{i-1}(g_{n-1}), \, \cdots ,\, d_0(g_{n-i}) \cdot g_{n-i-1}, \, g_{n - i - 2}, \, \cdots, \, g_0 \big) & \text{if} & i \lt n \\ \big( d_n(g_n), \, d_{n-1}(g_{n-1}), \, \cdots, \, d_1(g_1) \big) & \text{if} & i = n } \right. \end{aligned}
• degeneracy maps are given by

(2)\begin{aligned} & s_i(g_n, g_{n-1}, \cdots, g_0) \\ & \;\coloneqq\; \big( s_i(g_n), \, s_{i - 1}(g_{n-1}), \, \cdots, \, s_0(g_{n-i}), \, e, \, g_{n-i-1}, \, \cdots, \, g_0 \big) \,, \end{aligned}

where $e$ denotes the respective neutral element.

This carries a $G$-action by left multiplication on the top degree component:

(3)$\array{ G \times W G &\overset{}{\longrightarrow}& W G \\ \big(h_n, (g_n, g_{n-1}, \cdots, g_0)\big) &\mapsto& (h_n \cdot g_n, \, g_{n-1}, \cdots, g_0) \mathrlap{\,.} }$

###### Remark

It is the straightforward simplicial incarnation of the left $G$-action (3) that singles out the model $W G$ (Def. ) for the universal simplicial principal space. For another model with an alternative good property see at groupal model for universal principal simplicial complex.

###### Definition

(standard simplicial classifying complex)
For $G$ a simplicial group, its standard simplicial classifying complex is the quotient of $W G$ (Def. ) by its $G$-action (3)

(4)$\overline{W} G \;\coloneqq\; (W G) / G \,.$

The corresponding quotient coprojection, whose fiber is, manifestly, $G$

(5)$\array{ G &\xhookrightarrow{fib}& W G \\ && \big\downarrow \\ && \overline{W} G \mathrlap{\; \coloneqq (W G)/G} }$

is known as the standard model for the simplicial $G$-universal principal bundle (see below).

This means, under the isomorphism

$\big( \overline{W}G \big)_{n} \;\coloneqq\; (W G)_n/G_n \;\simeq\; G_n/G_n \times G_{n-1} \times \cdots \times G_0 \;\simeq\; G_{n-1} \times \cdots \times G_{0} \,,$

that the above face maps (1) and degeneracy maps (2) of $W G$ imply the following structure maps on the simplicial classifying complex:

$\overline{W}G \;\;\; \in \; SimplicialSets$
• underlying sets are

$(\overline{W}G)_n \;\coloneqq\; G_{n-1} \times \cdots \times G_1 \times G_0 \,;$
• face maps are given by:

(6)\begin{aligned} & d_i \big( g_{n-1}, \cdots, g_0 \big) \\ & \;\coloneqq\; \left\{ \array{ \big( g_{n - 2}, \, \cdots, \, g_0 \big) & \text{if} & i = 0 \\ \big( d_{i-1}(g_{n-1}), \, \cdots ,\, d_0(g_{n-i}) \cdot g_{n-i-1}, \, g_{n - i - 2}, \, \cdots, \, g_0 \big) & \text{if} & 0 \lt i \lt n \\ \big( d_{n-1}(g_{n-1}), \, \cdots, \, d_1(g_1) \big) & \text{if} & i = n \mathrlap{\,;} } \right. \end{aligned}
• degeneracy maps are given by:

(7)\begin{aligned} & s_i(g_{n-1}, \cdots, g_0) \\ & \;\coloneqq\; \left\{ \array{ \big( e, \, g_{n-1}, \, \cdots, \, g_0 \big) & \text{if} & i = 0 \\ \big( s_{i - 1}(g_{n-1}), \, \cdots, \, s_0(g_{n-i}), \, e, \, g_{n-i-1}, \, \cdots, \, g_0 \big) & \text{if} & 0 \lt i \mathrlap{\,.} } \right. \end{aligned}

(This goes back to MacLane 1954, p. 3, Kan 1958, Def. 10.3; the above follows Goerss & Jardine 1999/2009, p. 269.)

###### Remark

(décalage)
Conversely, comparison of Def. with Def. shows that $W G$ is obtained from $\overline{W} G$ by shifting down in degree and discarding the 0th face- and degeneracy maps:

$(W G)_{n} \;\simeq\; (\overline{W}G)_{n+1}$
\begin{aligned} d^{W G}_i & \;=\; d^{ \overline{W}G }_{i+1} \\ s^{W G}_i & \;=\; s^{\overline{W}G}_{i + 1} \mathrlap{\,.} \end{aligned}

One refers to this relation as saying that $W G$ is the décalage of $\overline{W}G$, in the form

$W G \;=\; Dec^0\big( \overline{W}G \big) \,.$

See the example there.

###### Example

(low-dimension cells of $W G$)
Unwinding the definition of the face maps (1), one finds that the generic 1-simplex in $W G$ (Def. ) looks as follows:

while the generic 2-simplex in $W G$ looks as follows:

###### Example

(simplicial classifying space of an ordinary group)
In the special case that

$G \in Groups \overset{const}{\hookrightarrow} SimplicialGroups$

is an ordinary discrete group regarded as a simplicial group (which is constant as a functor on the opposite simplex category) the definitions in Def. reduce as follows:

The simplicial set $W G$ is that whose

• underlying sets are

$(W G)_n \;\coloneqq\; G^{\times_{n + 1}}$
• face maps are given by

$d_i \big( g_n, g_{n-1}, \cdots, g_0 \big) \;\coloneqq\; \left\{ \array{ \big( g_n, \, g_{n-1}, \, \cdots ,\, g_{n-i} \cdot g_{n-i-1}, \, g_{n - i - 2}, \, \cdots, \, g_0 \big) & \text{if} & i \lt n \\ \big( g_n, \, g_{n-1}, \, \cdots, \, g_1 \big) & \text{if} & i = n } \right.$
• degeneracy maps are given by

$s_i(g_n, g_{n-1}, \cdots, g_1) \;\coloneqq\; \big( g_n, \, g_{n-1}, \, \cdots, \, g_{n-1}, \, e, \, g_{n-i-1}, \, \cdots, \, g_0 \big) \,,$

This identifies

$W G \;=\; N \big( G \times G \rightrightarrows G \big)$

with the nerve of the action groupoid of $G$ acting on itself by right multiplication (isomorphic to the pair groupoid on the underlying set of $G$):

Finally this means that the simplicial classifying complex (4) of an ordinary group is isomorphic to the nerve of its delooping groupoid:

$\overline{W}G \;\simeq\; N \big( G \rightrightarrows \ast \big) \,.$

### Via total simplicial sets

Equivalently, $\overline{W}(-)$ is the following composite functor:

$\overline{W} \;\colon\; [\Delta^{op}, Groups] \overset {\;\;[\Delta^{op}, \mathbf{B}]\;\;} {\longrightarrow} [\Delta^{op}, Groupoids] \overset {\;\;[\Delta^{op}, N]\;\;} {\longrightarrow} [\Delta^{op}, SimplicialSets] \overset {\;\;\sigma_\ast\;\;} {\longrightarrow} SimplicialSets \,.$

(Stevenson 11, Lemma 15 following John Duskin, see also NSS 12, Def. 3.26)

Here:

• $\mathbf{B} \;\colon\; Groups \to Groupoids$ forms the one-object groupoid with hom-set the given group (the delooping groupoid);

• $N \;\colon\; Groupoids \longrightarrow SimplicialSets$ is the nerve construction;

• $\sigma_\ast$ is the total simplicial set-functor (right adjoint to pre-composition with ordinal sum).

## Properties

In all of the following, $G$ is any simplicial group.

### Basic properties

###### Proposition

(simplicial classifying spaces are Kan complexes)
The underlying simplicial set of any simplicial classifying $\overline{W}G$ (Def. ) is a Kan complex.

(e.g. Goerss & Jardine 09, Sec. V Cor. 6.8 (p. 287))
###### Proof

This follows as the combination of the following facts:

1. every simplicial group is fibrant in the projective model structure on simplicial sets (this Prop.);

2. $\overline{W}(-)$ is a right Quillen functor from there to the injective model structure on reduced simplicial sets (Prop. );

3. every injectively fibrant reduced simplicial set is a Kan complex (this Prop.).

###### Proposition

For a simplicial group action of $G$ on a Kan complex $X$, the canonical coprojection from the Borel construction to the simplicial classifying space is a Kan fibration:

$(W G \times X)/G \xrightarrow{ \;\in Fib \; } \overline{W}G \,.$

###### Proof

By the fact (this Prop.) that the Borel construction is a right Quillen functor from the model structure on simplicial group actions to the slice model structure of the classical model structure on simplicial sets over the simplicial classifying space; see this Example.

In particular:

###### Proposition

The coprojection $W G \overset{}{\longrightarrow} \overline{W}G$ (5) is a Kan fibration.

(e.g Goerss & Jardine 09, Sec. V Lemma 4.1 (p. 270))
###### Proof

This is the special case of Prop. for $X = G$ equipped with the left multiplication action on itself, using again that the underlying simplicial set of any simplicial group is a Kan complex (this Prop.):

$\array{ G &\longrightarrow& W G \\ && \big\downarrow{}^{\mathrlap{\in Fib}} \\ && \overline{W}G } {\phantom{AAAAAA}} \simeq {\phantom{AAAA}} \left( W G \times \left( \array{ G \\ \downarrow \\ \ast } \right) \right) \big/ G \,.$

###### Proposition

The simplicial set $W G$ is contractible.

(e.g Goerss & Jardine 09, Sec. V Lemma 4.6 (p. 270), see also the discussion at décalage)

###### Proposition

The simplicial homotopy groups of $\overline{W} G$ are those of $G$, shifted up in degree by one:

$\pi_n(G) \;\simeq\; \pi_{n+1}\big(\overline{W}(G)\big) \,.$

###### Proof

By Prop. the universal simplicial principal bundle (5) is a Kan fibration between Kan complexes (by Prop. and this Prop.). Therefore this is a homotopy fiber sequence (by this Prop.)

$G \xrightarrow{hofib} W G \xrightarrow{\;\;q\;\;} \overline{W}G \,.$

This implies a long exact sequence of homotopy groups of the form

$\cdots \xrightarrow{\;} \pi_{n+1}(W G) \xrightarrow{\;\;} \pi_{n+1}(\overline{W}G) \xrightarrow{\;\;} \pi_n(G) \xrightarrow{\;\;} \pi_n(W G) \xrightarrow{\;} \cdots \,.$

But Prop. says that $\pi_n(W G)$ is trivial for all $n$, so that this collapses to short exact sequences:

$0 \xrightarrow{\;\;} \pi_{n+1}(\overline{W}G) \xrightarrow{\;\simeq\;} \pi_n(G) \xrightarrow{\;\;} 0$

which exhibit the claim to be proven.

###### Proposition

Let $\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2$ be a homomorphism of simplicial groups which is a Kan fibration. Then the induced morphism of simplicial classifying spaces $\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2$ is a Kan fibration if and only if $\phi$ is a surjection on connected components: $\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1)$.

(Goerss & Jardine, Ch. V, Cor. 6.9, see the proof here)

### Classification of simplicial principal bundles

The object $\overline{W}G$ serves as the classifying space for simplicial principal bundles (May 67, §21, Goerss & Jardine 09, Section V, Thm. 3.9, see also NSS 12, Section 4.1).

### Quillen equivalence between simplicial groups and reduced simplicial sets

###### Proposition

(Quillen equivalence between simplicial groups and reduced simplicial sets)
The simplicial classifying space-construction $\overline{W}(-)$ (Def. ) is the right adjoint in a Quillen equivalence between the projective model structure on simplicial groups and the injective model structure on reduced simplicial sets.

$Groups(sSet)_{proj} \underoverset {\underset{\overline{W}}{\longrightarrow}} {\overset{ \Omega }{\longleftarrow}} {\;\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;\;} (sSet_{\geq 0})_{inj} \,.$

The left adjoint $\Omega$ is the simplicial loop space-construction.

### Slice model structure

The slice model category of the classical model structure on simplicial sets over the simplicial classifying complex $\overline{W}G$ is Quillen equivalent to the Borel model structure for $G$-equivariant homotopy theory:

$\big( SimplicialSets_{Qu} \big)_{/\overline{W} G} \;\simeq_{Qu}\; G Actions(SimplicialSets)_{proj} \,.$

(Dror, Dwyer & Kan 1980) See there for details. This is a model for the general abstract situation discussed at ∞-action.

## References

The idea of constructing $\overline{W}$ using the bar construction is due to Eilenberg and MacLane, who apply it to simplicial rings with the usual tensor product operation:

This was also later discussed in

• Saunders MacLane, Constructions simpliciales acycliques, Colloque Henri Poincaré 1954 (pdf) (See, in particular, §3.)

The first reference where $\overline{W}$ is defined explicitly for simplicial groups and the adjunction between simplicial groups and reduced simplicial sets is explicitly spelled out is

• Daniel Kan, Sections 10-11 in: On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958), 38-53 (jstor:1970042)

The left adjoint simplicial loop space functor $L$ is also discussed by Kan (there denoted “$G$”) in

• Daniel M. Kan, §7 of: A combinatorial definition of homotopy groups, Annals of Mathematics 67:2 (1958), 282–312. doi.

The Quillen equivalence was established in

• Dan Quillen, Section 2 of: Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (jstor:1970725)

Textbook accounts:

Streamlining:

Identification of the slice model structure over $\overline{W}G$ with the Borel model structure:

Generalization to simplicial presheaves:

Last revised on July 5, 2021 at 08:48:12. See the history of this page for a list of all contributions to it.